CINXE.COM
A001844 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A001844 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A001844" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA001844">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A001844 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A001844 </div> <div class=seqname> Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z; then sequence gives Z values. <br><font size=-1>(Formerly M3826 N1567)</font> </div> </div> <div class=scorerefs> 334 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513</div> <div class=seqdatalinks> (<a href="/A001844/list">list</a>; <a href="/A001844/graph">graph</a>; <a href="/search?q=A001844+-id:A001844">refs</a>; <a href="/A001844/listen">listen</a>; <a href="/history?seq=A001844">history</a>; <a href="/search?q=id:A001844&fmt=text">text</a>; <a href="/A001844/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>These are Hogben's central polygonal numbers denoted by</div> <div class=sectline>...2...</div> <div class=sectline>....P..</div> <div class=sectline>...4.n.</div> <div class=sectline>Numbers of the form (k^2+1)/2 for k odd.</div> <div class=sectline>(y(2x+1))^2 + (y(2x^2+2x))^2 = (y(2x^2+2x+1))^2. E.g., let y = 2, x = 1; (2(2+1))^2 + (2(2+2))^2 = (2(2+2+1))^2, (2(3))^2 + (2(4))^2 = (2(5))^2, 6^2 + 8^2 = 10^2, 36 + 64 = 100. - Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 08 2002</div> <div class=sectline>a(n) is also the number of 3 X 3 magic squares with sum 3(n+1). - Sharon Sela (sharonsela(AT)hotmail.com), May 11 2002</div> <div class=sectline>For n > 0, a(n) is the smallest k such that zeta(2) - Sum_{i=1..k} 1/i^2 <= zeta(3) - Sum_{i=1..n} 1/i^3. - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, May 17 2002</div> <div class=sectline>Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.</div> <div class=sectline>The prime terms are given by <a href="/A027862" title="Primes of the form j^2 + (j+1)^2.">A027862</a>. - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, Jul 09 2004</div> <div class=sectline>First difference of a(n) is 4n = <a href="/A008586" title="Multiples of 4.">A008586</a>(n). Any entry k of the sequence is followed by k + 2*(1 + sqrt(2k - 1)). - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, Jun 04 2006</div> <div class=sectline>Integers of the form 1 + x + x^2/2 (generating polynomial is Schur's polynomial as in <a href="/A127876" title="Integers of the form (x^3)/6 + (x^2)/2 + x + 1.">A127876</a>). - <a href="/wiki/User:Artur_Jasinski">Artur Jasinski</a>, Feb 04 2007</div> <div class=sectline>If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 4-subsets of X intersecting both Y and Z. - <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, Aug 26 2007</div> <div class=sectline>Row sums of triangle <a href="/A132778" title="Triangle read by rows, n-1 terms of (2n - 1) followed by n.">A132778</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Sep 02 2007</div> <div class=sectline>Binomial transform of [1, 4, 4, 0, 0, 0, ...]; = inverse binomial transform of <a href="/A001788" title="a(n) = n*(n+1)*2^(n-2).">A001788</a>: (1, 6, 24, 80, 240, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Sep 02 2007</div> <div class=sectline>Narayana transform (<a href="/A001263" title="Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.">A001263</a>) of [1, 4, 0, 0, 0, ...]. Equals <a href="/A128064" title="Triangle T with T(n,n)=n, T(n,n-1)=-(n-1) and otherwise T(n,k)=0; 0<k<=n.">A128064</a> (unsigned) * [1, 2, 3, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 29 2007</div> <div class=sectline>n such that the Diophantine equation x^3 - y^3 = x*y + n has a solution with y = x-1. If that solution is (x,y) = (m+1,m) then m^2 + (m+1)^2 = n. Note that this Diophantine equation is an elliptic curve and (m+1,m) is an integer point on it. - <a href="/wiki/User:James_R._Buddenhagen">James R. Buddenhagen</a>, Aug 12 2008</div> <div class=sectline>Numbers n such that (n, n, 2*n-2) are the sides of an isosceles triangle with integer area. Also, n such that 2*n-1 is a square. - <a href="/wiki/User:James_R._Buddenhagen">James R. Buddenhagen</a>, Oct 17 2008</div> <div class=sectline>a(n) is also the least weight of self-conjugate partitions having n+1 different odd parts. - <a href="/wiki/User:Augustine_O._Munagi">Augustine O. Munagi</a>, Dec 18 2008</div> <div class=sectline>Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) = <a href="/A153869" title="Triangle read by rows, A129186 * A128064(unsigned).">A153869</a> * (1, 2, 3, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jan 03 2009</div> <div class=sectline>Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) where a(n) = 2n*(n-1)+1, all tuples of square numbers (X-Y, X, X+Y) are produced by ((m*(a(n)-2n))^2, (m*a(n))^2, (m*(a(n)+2n-2))^2) where m is a whole number. - <a href="/wiki/User:Doug_Bell">Doug Bell</a>, Feb 27 2009</div> <div class=sectline>Equals (1, 2, 3, ...) convolved with (1, 3, 4, 4, 4, ...). a(3) = 25 = (1, 2, 3, 4) dot (4, 4, 3, 1) = (4 + 8 + 9 + 4). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, May 01 2009</div> <div class=sectline>The running sum of squares taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009</div> <div class=sectline>Equals the odd integers convolved with (1, 2, 2, 2, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, May 25 2009</div> <div class=sectline>Equals the triangular numbers convolved with [1, 2, 1, 0, 0, 0, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a> & <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, May 29 2009</div> <div class=sectline>When the positive integers are written in a square array by diagonals as in <a href="/A038722" title="Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,2,3,4,... .">A038722</a>, a(n) gives the numbers appearing on the main diagonal. - <a href="/wiki/User:Joshua_Zucker">Joshua Zucker</a>, Jul 07 2009</div> <div class=sectline>The finite continued fraction [n,1,1,n] = (2n+1)/(2n^2 + 2n + 1) = (2n+1)/a(n); and the squares of the first two denominators of the convergents = a(n). E.g., the convergents and value of [4,1,1,4] = 1/4, 1/5, 2/9, 9/41 where 4^2 + 5^2 = 41. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jul 15 2010</div> <div class=sectline>From <a href="/wiki/User:Keith_Tyler">Keith Tyler</a>, Aug 10 2010: (Start)</div> <div class=sectline>Running sum of <a href="/A008574" title="a(0) = 1, thereafter a(n) = 4n.">A008574</a>.</div> <div class=sectline>Square open pyramidal number; that is, the number of elements in a square pyramid of height (n) with only surface and no bottom nodes. (End)</div> <div class=sectline>For k>0, x^4 + x^2 + k factors over the integers iff sqrt(k) is in this sequence. - <a href="/wiki/User:James_R._Buddenhagen">James R. Buddenhagen</a>, Aug 15 2010</div> <div class=sectline>Create the simple continued fraction from Pythagorean triples to get [2n + 1; 2n^2 + 2n,2n^2 + 2n + 1]; its value equals the rational number 2n +1 + a(n) / (4*n^4 + 8*n^3 + 6*n^2 + 2*n + 1). - <a href="/wiki/User:J._M._Bergot">J. M. Bergot</a>, Sep 30 2011</div> <div class=sectline>a(n), n >= 1, has in its prime number factorization only primes of the form 4*k+1, i.e., congruent 1 (mod 4) (see <a href="/A002144" title="Pythagorean primes: primes of the form 4*k + 1.">A002144</a>). This follows from the fact that a(n) is a primitive sum of two squares and odd. See Theorem 3.20, p. 164, in the given Niven-Zuckerman-Montgomery reference. E.g., a(3) = 25 = 5^2, a(6) = 85 = 5*17. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Mar 08 2012</div> <div class=sectline>From <a href="/wiki/User:Ant_King">Ant King</a>, Jun 15 2012: (Start)</div> <div class=sectline>a(n) is congruent to 1 (mod 4) for all n.</div> <div class=sectline>The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 5, 4, 7, 5, 7, 4, 5, 1.</div> <div class=sectline>The units' digits of the a(n) form a purely periodic palindromic 5-cycle 1, 5, 3, 5, 1.</div> <div class=sectline>(End)</div> <div class=sectline>Number of integer solutions (x,y) of |x| + |y| <= n. Geometrically: number of lattice points inside a square with vertex (n,0), (0,-n), (-n,0), (0,n). - <a href="/wiki/User:C茅sar_Eliud_Lozada">C茅sar Eliud Lozada</a>, Sep 18 2012</div> <div class=sectline>(a(n)-1)/a(n) = 2*x / (1+x^2) where x = (n-1)/n. Note that in this form, this is the velocity-addition formula according to the special theory of relativity (two objects traveling at 1/n slower than c relative to each other appear to travel at 1/a(n) less than c to a stationary observer). - <a href="/wiki/User:Christian_N._K._Anderson">Christian N. K. Anderson</a>, May 20 2013</div> <div class=sectline>A geometric curiosity: the envelope of the circles x^2 + (y-a(n)/2)^2 = ((2n+1)/2)^2 is the parabola y = x^2, the n=0 circle being the osculating circle at the parabola vertex. - <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Dec 02 2013</div> <div class=sectline>Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of internal regions into which the plane is divided (cf. <a href="/A051890" title="a(n) = 2*(n^2 - n + 1).">A051890</a>, <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>); a(n-1) = <a href="/A051890" title="a(n) = 2*(n^2 - n + 1).">A051890</a>(n) - 1 = <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n-1) - 2. - <a href="/wiki/User:Jaroslav_Krizek">Jaroslav Krizek</a>, Dec 27 2013</div> <div class=sectline>a(n) is also, of course, the scalar product of the 2-vector (n, n+1) (or (n+1, n)) with itself. The unique inverse of (n, n+1) as vector in the Clifford algebra over the Euclidean 2-space is (1/a(n))(0, n, n+1, 0) (similarly for the other vector). In general the unique inverse of such a nonzero vector v (odd element in Cl_2) is v^(-1) = (1/|v|^2) v. Note that the inverse with respect to the scalar product is not unique for any nonzero vector. See the P. Lounesto reference, sects. 1.7 - 1.12, pp. 7-14. See also the Oct 15 2014 comment in <a href="/A147973" title="a(n) = -2*n^2 + 12*n - 14.">A147973</a>. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Nov 06 2014</div> <div class=sectline>Subsequence of <a href="/A004431" title="Numbers that are the sum of 2 distinct nonzero squares.">A004431</a>, for n >= 1. - <a href="/wiki/User:Bob_Selcoe">Bob Selcoe</a>, Mar 23 2016</div> <div class=sectline>Numbers n such that 2n - 1 is a perfect square. - <a href="/wiki/User:Juri-Stepan_Gerasimov">Juri-Stepan Gerasimov</a>, Apr 06 2016</div> <div class=sectline>The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 574", based on the 5-celled von Neumann neighborhood. - <a href="/wiki/User:Robert_Price">Robert Price</a>, May 13 2016</div> <div class=sectline>a(n) is the first integer in a sum of (2*n + 1)^2 consecutive integers that equals (2*n + 1)^4. - <a href="/wiki/User:Patrick_J._McNab">Patrick J. McNab</a>, Dec 24 2016</div> <div class=sectline>Central elements of odd-length rows of the triangular array of positive integers. a(n) is the mean of the numbers in the (2*n + 1)-th row of this triangle. - <a href="/wiki/User:David_James_Sycamore">David James Sycamore</a>, Aug 01 2018</div> <div class=sectline>Intersection of <a href="/A000982" title="a(n) = ceiling(n^2/2).">A000982</a> and <a href="/A080827" title="Rounded up staircase on natural numbers.">A080827</a>. - <a href="/wiki/User:David_James_Sycamore">David James Sycamore</a>, Aug 07 2018</div> <div class=sectline>An off-diagonal of the array of Delannoy numbers, <a href="/A008288" title="Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.">A008288</a>, (or a row/column when the array is shown as a square). As such, this is one of the crystal ball sequences. - <a href="/wiki/User:Jack_W_Grahl">Jack W Grahl</a>, Feb 15 2021 and <a href="/wiki/User:Shel_Kaphan">Shel Kaphan</a>, Jan 18 2023</div> <div class=sectline>a(n) appears as a solution to a "Riddler Express" puzzle on the FiveThirtyEight website. The Jan 21 2022 issue (problem) and the Jan 28 2022 issue (solution) present the following puzzle and include a proof. - Fold a square piece of paper in half, obtaining a rectangle. Fold again to obtain a square with 1/4 the size of the original square. Then make n cuts through the folded paper. a(n) is the greatest number of pieces of the unfolded paper after the cutting. - <a href="/wiki/User:Manfred_Boergens">Manfred Boergens</a>, Feb 22 2022</div> <div class=sectline>a(n) is (1/6) times the number of 2 X 2 triangles in the n-th order hexagram with 12*n^2 cells. - <a href="/wiki/User:Donghwi_Park">Donghwi Park</a>, Feb 06 2024</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.</div> <div class=sectline>A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.</div> <div class=sectline>L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.</div> <div class=sectline>John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.</div> <div class=sectline>Pertti Lounesto, Clifford Algebras and Spinors, second edition, Cambridge University Press, 2001.</div> <div class=sectline>S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.</div> <div class=sectline>Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>Travers et al., The Mysterious Lost Proof, Using Advanced Algebra, (1976), pp. 27.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A001844/b001844.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>M. Ahmed, J. De Loera and R. Hemmecke, <a href="http://arxiv.org/abs/math/0201108">Polyhedral Cones of Magic Cubes and Squares</a>, arXiv:math/0201108 [math.CO], 2002.</div> <div class=sectline>U. Alfred, <a href="http://www.jstor.org/stable/2688938">n and n+1 consecutive integers with equal sums of squares</a>, Math. Mag., 35 (1962), 155-164.</div> <div class=sectline>Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.</div> <div class=sectline>Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.</div> <div class=sectline>Matthias Beck, Moshe Cohen, Jessica Cuomo, and Paul Gribelyuk, <a href="https://arxiv.org/abs/math/0201013">The number of "magic" squares and hypercubes</a>, arXiv:math/0201013 [math.CO], 2002-2005.</div> <div class=sectline>Arthur T. Benjamin and Doron Zeilberger, <a href="http://www.emis.de/journals/INTEGERS/papers/f30/f30.Abstract.html">Pythagorean Primes and Palindromic Continued Fractions</a>, Electronic Journal of Combinatorial Number Theory, 5(1) 2005, #A30.</div> <div class=sectline>J. A. De Loera, D. C. Haws and M. Koppe, <a href="http://arxiv.org/abs/0710.4346">Ehrhart Polynomials of Matroid Polytopes and Polymatroids</a>, arXiv:0710.4346 [math.CO], 2007; Discrete Comput. Geom., 42 (2009), 670-702.</div> <div class=sectline>FiveThirtyEight, <a href="https://fivethirtyeight.com/features/can-you-tune-up-the-truck/">"Riddler Express" paper cutting problem and solution</a>, Jan 28 2022.</div> <div class=sectline>D. C. Haws, <a href="http://www.math.ucdavis.edu/~haws/Matroids/">Matroids</a> [Broken link, Oct 30 2017]</div> <div class=sectline>D. C. Haws, <a href="https://www.math.ucdavis.edu/~mkoeppe/art/Matroids/">Matroids</a> [Copy on website of Matthias Koeppe]</div> <div class=sectline>D. C. Haws, <a href="/A160747/a160747.pdf">Matroids</a> [Cached copy, pdf file only]</div> <div class=sectline>L. Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n25">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, pp. 22 and 36.</div> <div class=sectline>Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>. [Broken link; <a href="https://web.archive.org/web/20110204173116/http://www.pmfbl.org:80/janjic/">WayBackMachine archive</a>.]</div> <div class=sectline>Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.html">On a class of polynomials with integer coefficients</a>, JIS 11 (2008) 08.5.</div> <div class=sectline>Milan Janji膰, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.</div> <div class=sectline>Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.</div> <div class=sectline>Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.</div> <div class=sectline>Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>.</div> <div class=sectline>G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hi茅rarchies de segments</a>, Cahiers Bureau Universitaire Recherche Op茅rationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.</div> <div class=sectline>G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hi茅rarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Op茅rationnelle, Institut de Statistique, Universit茅 de Paris, #20 (1973). (Annotated scanned copy)</div> <div class=sectline>A. O. Munagi, <a href="http://dx.doi.org/10.1016/j.disc.2007.05.022">Pairing conjugate partitions by residue classes</a>, Discrete Math., 308 (2008), 2492-2501.</div> <div class=sectline>Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, <a href="http://arxiv.org/abs/1511.00080">Pattern Avoidance in Task-Precedence Posets</a>, arXiv:1511.00080 [math.CO], 2015-2016.</div> <div class=sectline>Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de s茅ries g茅n茅ratrices et quelques conjectures</a>, Dissertation, Universit茅 du Qu茅bec 脿 Montr茅al, 1992; arXiv:0911.4975 [math.NT], 2009.</div> <div class=sectline>Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992</div> <div class=sectline>John A. Jr. Rochowicz, <a href="http://epublications.bond.edu.au/ejsie/vol8/iss2/4">Harmonic Numbers: Insights, Approximations and Applications</a>, Spreadsheets in Education (eJSiE), 2015, Vol. 8: Iss. 2, Article 4.</div> <div class=sectline>Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3252339">Groupoid of OEIS A001844 Numbers (centered square numbers)</a>, Politecnico di Torino, Italy (2019).</div> <div class=sectline>R. G. Stanton and D. D. Cowan, <a href="http://dx.doi.org/10.1137/1012049">Note on a "square" functional equation</a>, SIAM Rev., 12 (1970), 277-279.</div> <div class=sectline>David James Sycamore, <a href="/A001844/a001844.jpg">Triangular array</a></div> <div class=sectline>Leo Tavares, <a href="/A001844/a001844_1.jpg">Illustration: Diamond Rows</a></div> <div class=sectline>B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>, <a href="http://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>, <a href="http://mathworld.wolfram.com/Diamond.html">Diamond</a>, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>, and <a href="http://mathworld.wolfram.com/vonNeumannNeighborhood.html">von Neumann Neighborhood</a>.</div> <div class=sectline><a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a></div> <div class=sectline><a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.</div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).</div> <div class=sectline>Nicolay Avilov, <a href="/A001844/a001844_2.jpg">Graphical representation of the sequence members</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = 2*n^2 + 2*n + 1 = n^2 + (n+1)^2.</div> <div class=sectline>a(n) = 1 + 3 + 5 + ... + 2*n-1 + 2*n+1 + 2*n-1 + ... + 3 + 1. - <a href="/wiki/User:Amarnath_Murthy">Amarnath Murthy</a>, May 28 2001</div> <div class=sectline>a(n) = 1/real(z(n+1)) where z(1)=i, (i^2=-1), z(k+1) = 1/(z(k)+2i). - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Aug 06 2002</div> <div class=sectline>Nearest integer to 1/Sum_{k>n} 1/k^3. - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Jun 12 2003</div> <div class=sectline>G.f.: (1+x)^2/(1-x)^3.</div> <div class=sectline>E.g.f.: exp(x)*(1+4x+2x^2).</div> <div class=sectline>a(n) = a(n-1) + 4n.</div> <div class=sectline>a(-n) = a(n-1).</div> <div class=sectline>a(n) = <a href="/A064094" title="Triangle composed of generalized Catalan numbers.">A064094</a>(n+3, n) (fourth diagonal).</div> <div class=sectline>a(n) = 1 + Sum_{j=0..n} 4*j. - Xavier Acloque, Oct 08 2003</div> <div class=sectline>a(n) = <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n)+1 = (<a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n)+1)/2. - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, May 25 2004</div> <div class=sectline>a(n) = Sum_{k=0..n+1} (-1)^k*binomial(n, k)*Sum_{j=0..n-k+1} binomial(n-k+1, j)*j^2. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Dec 22 2004</div> <div class=sectline>a(n) = ceiling((2n+1)^2/2). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Jul 16 2006</div> <div class=sectline>a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=1, a(1)=5, a(2)=13. - <a href="/wiki/User:Jaume_Oliver_Lafont">Jaume Oliver Lafont</a>, Dec 02 2008</div> <div class=sectline>a(n)*a(n-1) = 4*n^4 + 1 for n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 12 2009</div> <div class=sectline>Prefaced with a "1" (1, 1, 5, 13, 25, 41, ...): a(n) = 2*n*(n-1)+1. - <a href="/wiki/User:Doug_Bell">Doug Bell</a>, Feb 27 2009</div> <div class=sectline>a(n) = sqrt((<a href="/A056220" title="a(n) = 2*n^2 - 1.">A056220</a>(n)^2 + <a href="/A056220" title="a(n) = 2*n^2 - 1.">A056220</a>(n+1)^2) / 2). - <a href="/wiki/User:Doug_Bell">Doug Bell</a>, Mar 08 2009</div> <div class=sectline>a(n) = floor(2*(n+1)^3/(n+2)). - <a href="/wiki/User:Gary_Detlefs">Gary Detlefs</a>, May 20 2010</div> <div class=sectline>a(n) = <a href="/A000330" title="Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.">A000330</a>(n) - <a href="/A000330" title="Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.">A000330</a>(n-2). - <a href="/wiki/User:Keith_Tyler">Keith Tyler</a>, Aug 10 2010</div> <div class=sectline>a(n) = <a href="/A069894" title="Centered square numbers: a(n) = 4*n^2 + 4*n + 2.">A069894</a>(n)/2. - <a href="/wiki/User:J._M._Bergot">J. M. Bergot</a>, Jun 11 2012</div> <div class=sectline>a(n) = 2*a(n-1) - a(n-2) + 4. - <a href="/wiki/User:Ant_King">Ant King</a>, Jun 12 2012</div> <div class=sectline>Sum_{n>=0} 1/a(n) = Pi/2*tanh(Pi/2) = 1.4406595199775... = <a href="/A228048" title="Decimal expansion of (Pi/2)*tanh(Pi/2).">A228048</a>. - <a href="/wiki/User:Ant_King">Ant King</a>, Jun 15 2012</div> <div class=sectline>a(n) = <a href="/A209297" title="Triangle read by rows: T(n,k) = k*n + k - n, 1 <= k <= n.">A209297</a>(2*n+1,n+1). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 19 2013</div> <div class=sectline>a(n)^3 = <a href="/A048395" title="Sum of consecutive nonsquares.">A048395</a>(n)^2 + <a href="/A048395" title="Sum of consecutive nonsquares.">A048395</a>(-n-1)^2. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jan 19 2013</div> <div class=sectline>a(n) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(2n+1) - n. - <a href="/wiki/User:Ivan_N._Ianakiev">Ivan N. Ianakiev</a>, Nov 08 2013</div> <div class=sectline>a(n) = <a href="/A251599" title="Centers of rows of the triangular array formed by the natural numbers.">A251599</a>(3*n+1). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 13 2014</div> <div class=sectline>a(n) = <a href="/A101321" title="Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.">A101321</a>(4,n). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 28 2016</div> <div class=sectline>From <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Jul 30 2016: (Start)</div> <div class=sectline>a(n) = Sum_{k=0..n} <a href="/A008574" title="a(0) = 1, thereafter a(n) = 4n.">A008574</a>(k).</div> <div class=sectline>Sum_{n>=0} (-1)^(n+1)*a(n)/n! = exp(-1) = <a href="/A068985" title="Decimal expansion of 1/e.">A068985</a>. (End)</div> <div class=sectline>a(n) = 4 * <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) + 1. - <a href="/wiki/User:Bruce_J._Nicholson">Bruce J. Nicholson</a>, Jul 10 2017</div> <div class=sectline>a(n) = <a href="/A002522" title="a(n) = n^2 + 1.">A002522</a>(n) + <a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a>(n) = <a href="/A002522" title="a(n) = n^2 + 1.">A002522</a>(n+1) + <a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a>(n-1). - <a href="/wiki/User:Bruce_J._Nicholson">Bruce J. Nicholson</a>, Aug 05 2017</div> <div class=sectline>Sum_{n>=0} a(n)/n! = 7*e. Sum_{n>=0} 1/a(n) = <a href="/A228048" title="Decimal expansion of (Pi/2)*tanh(Pi/2).">A228048</a>. - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jun 20 2020</div> <div class=sectline>a(n) = <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a>(n+1) + <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n-1). - <a href="/wiki/User:Charlie_Marion">Charlie Marion</a>, Nov 16 2020</div> <div class=sectline>a(n) = Integral_{x=0..2n+2} |1-x| dx. - <a href="/wiki/User:Pedro_Caceres">Pedro Caceres</a>, Dec 29 2020</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 17 2021: (Start)</div> <div class=sectline>Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(Pi/2).</div> <div class=sectline>Product_{n>=1} (1 - 1/a(n)) = Pi*csch(Pi)*sinh(Pi/2). (End)</div> <div class=sectline>a(n) = <a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n+1) + 1 - <a href="/A028242" title="Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8.">A028242</a>(n). - <a href="/wiki/User:Charlie_Marion">Charlie Marion</a>, Apr 05 2022</div> <div class=sectline>a(n) = <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n) - <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n). - <a href="/wiki/User:Leo_Tavares">Leo Tavares</a>, Sep 16 2022</div> <div class=sectline>For n>0, a(n) = <a href="/A101096" title="Third differences of fifth powers (A000584).">A101096</a>(n+2) / 30. - <a href="/wiki/User:Andy_Nicol">Andy Nicol</a>, Feb 06 2025</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>G.f. = 1 + 5*x + 13*x^2 + 25*x^3 + 41*x^4 + 61*x^5 + 85*x^6 + 113*x^7 + 145*x^8 + ...</div> <div class=sectline>The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...</div> <div class=sectline>The first four such partitions, corresponding to a(n) = 0,1,2,3, are 1, 3+1+1, 5+3+3+1+1, 7+5+5+3+3+1+1. - <a href="/wiki/User:Augustine_O._Munagi">Augustine O. Munagi</a>, Dec 18 2008</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>:=-(z+1)**2/(z-1)**3; # <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[2n(n + 1) + 1, {n, 0, 50}]</div> <div class=sectline>FoldList[#1 + #2 &, 1, 4 Range@ 50] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Feb 02 2011 *)</div> <div class=sectline>maxn := 47; Flatten[Table[SeriesCoefficient[Series[(n + (n - 1)*x)/(1 - x)^2, {x, 0, maxn}], k], {n, maxn}, {k, n - 1, n - 1}]] (* <a href="/wiki/User:L._Edson_Jeffery">L. Edson Jeffery</a>, Aug 24 2014 *)</div> <div class=sectline>CoefficientList[ Series[-(x^2 + 2x + 1)/(x - 1)^3, {x, 0, 48}], x] (* or *)</div> <div class=sectline>LinearRecurrence[{3, -3, 1}, {1, 5, 13}, 48] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Aug 01 2018 *)</div> <div class=sectline>Total/@Partition[Range[0, 50]^2, 2, 1] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Dec 05 2020 *)</div> <div class=sectline>Table[ j! Coefficient[Series[Exp[x]*(1 + 4*x + 2*x^2), {x, 0, 20}], x,</div> <div class=sectline>j], {j, 0, 20}] (* <a href="/wiki/User:Nikolaos_Pantelidis">Nikolaos Pantelidis</a>, Feb 07 2023 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n) = 2*n*(n+1) + 1};</div> <div class=sectline>(PARI) x='x+O('x^200); Vec((1+x)^2/(1-x)^3) \\ <a href="/wiki/User:Altug_Alkan">Altug Alkan</a>, Mar 23 2016</div> <div class=sectline>(Sage) [i**2 + (i + 1)**2 for i in range(46)] # <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Jun 27 2008</div> <div class=sectline>(Haskell)</div> <div class=sectline>a001844 n = 2 * n * (n + 1) + 1</div> <div class=sectline>a001844_list = zipWith (+) a000290_list $ tail a000290_list</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 04 2012</div> <div class=sectline>(Magma) [2*n^2 + 2*n + 1: n in [0..50]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jan 19 2013</div> <div class=sectline>(Magma) [n: n in [0..4400] | IsSquare(2*n-1)]; // <a href="/wiki/User:Juri-Stepan_Gerasimov">Juri-Stepan Gerasimov</a>, Apr 06 2016</div> <div class=sectline>(Python) print([2*n*(n+1)+1 for n in range(48)]) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Jan 05 2021</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>X values are <a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>; Y values are <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>.</div> <div class=sectline>Cf. <a href="/A008586" title="Multiples of 4.">A008586</a> (first differences), <a href="/A005900" title="Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.">A005900</a> (partial sums), <a href="/A254373" title="Digital roots of centered square numbers (A001844).">A254373</a> (digital root).</div> <div class=sectline>Cf. <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>, <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A001263" title="Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.">A001263</a>, <a href="/A001788" title="a(n) = n*(n+1)*2^(n-2).">A001788</a>, <a href="/A002061" title="Central polygonal numbers: a(n) = n^2 - n + 1.">A002061</a>, <a href="/A004431" title="Numbers that are the sum of 2 distinct nonzero squares.">A004431</a> (numbers that are the sum of 2 distinct nonzero squares), <a href="/A005448" title="Centered triangular numbers: a(n) = 3*n*(n-1)/2 + 1.">A005448</a>, <a href="/A005891" title="Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.">A005891</a>, <a href="/A008844" title="Squares of sequence A001653: y^2 such that x^2 - 2*y^2 = -1 for some x.">A008844</a> (terms which are perfect squares), <a href="/A048395" title="Sum of consecutive nonsquares.">A048395</a>, <a href="/A051890" title="a(n) = 2*(n^2 - n + 1).">A051890</a>, <a href="/A056106" title="Second spoke of a hexagonal spiral.">A056106</a>, <a href="/A101096" title="Third differences of fifth powers (A000584).">A101096</a>, <a href="/A127876" title="Integers of the form (x^3)/6 + (x^2)/2 + x + 1.">A127876</a>, <a href="/A128064" title="Triangle T with T(n,n)=n, T(n,n-1)=-(n-1) and otherwise T(n,k)=0; 0<k<=n.">A128064</a>, <a href="/A132778" title="Triangle read by rows, n-1 terms of (2n - 1) followed by n.">A132778</a>, <a href="/A147973" title="a(n) = -2*n^2 + 12*n - 14.">A147973</a>, <a href="/A153869" title="Triangle read by rows, A129186 * A128064(unsigned).">A153869</a>, <a href="/A240876" title="Expansion of (1 + x)^11 / (1 - x)^12.">A240876</a>, <a href="/A251599" title="Centers of rows of the triangular array formed by the natural numbers.">A251599</a> <a href="/A000982" title="a(n) = ceiling(n^2/2).">A000982</a>, <a href="/A080827" title="Rounded up staircase on natural numbers.">A080827</a>, <a href="/A008288" title="Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.">A008288</a>.</div> <div class=sectline>Right edge of <a href="/A055096" title="Triangle read by rows, sums of 2 distinct nonzero squares: T(n,k) = k^2+n^2, (n>=2, 1 <= k <= n-1)">A055096</a>; main diagonal of <a href="/A069480" title="Determinant of rank n matrix of 1..n^2 filled successively along antidiagonals.">A069480</a>, <a href="/A078475" title="Determinant of rank n matrix of 1..n^2 filled successively back and forth along antidiagonals.">A078475</a>, <a href="/A129312" title="A minimal 2 X 2 subdeterminant array.">A129312</a>.</div> <div class=sectline>Row n=2 (or column k=2) of <a href="/A008288" title="Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.">A008288</a>.</div> <div class=sectline>Cf. <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>.</div> <div class=sectline>Sequence in context: <a href="/A081961" title="Hypotenuses of primitive Pythagorean triangles with an odd short leg (or an even long leg) sorted on semiperimeter.">A081961</a> <a href="/A096891" title="Least hypotenuse of primitive Pythagorean triangles with odd leg 2n+1.">A096891</a> <a href="/A380280" title="Values corresponding to ties in A377091, in order of appearance.">A380280</a> * <a href="/A099776" title="Length of the hypotenuse of an integer right triangle with the hypotenuse being one more than the longer side. The shorter s...">A099776</a> <a href="/A301302" title="Partial sums of A301301.">A301302</a> <a href="/A133322" title="Centered square numbers that are prime powers of the form (4n+1)^k.">A133322</a></div> <div class=sectline>Adjacent sequences: <a href="/A001841" title="Related to Zarankiewicz's problem.">A001841</a> <a href="/A001842" title="Expansion of Sum_{n>=0} x^(4*n+3)/(1 - x^(4*n+3)).">A001842</a> <a href="/A001843" title="The coding-theoretic function A(n,4,4).">A001843</a> * <a href="/A001845" title="Centered octahedral numbers (crystal ball sequence for cubic lattice).">A001845</a> <a href="/A001846" title="Centered 4-dimensional orthoplex numbers (crystal ball sequence for 4-dimensional cubic lattice).">A001846</a> <a href="/A001847" title="Crystal ball sequence for 5-dimensional cubic lattice.">A001847</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Partially edited by <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Mar 11 2010</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified March 13 19:40 EDT 2025. Contains 381739 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>