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A016813 - OEIS
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We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).</p> </div> <div class="donate"> <div id="donate-button-container"> <div id="donate-button"></div> <script src="https://www.paypalobjects.com/donate/sdk/donate-sdk.js" charset="UTF-8"></script> <script> PayPal.Donation.Button({ env:'production', hosted_button_id:'SVPGSDDCJ734A', image: { src:'https://www.paypalobjects.com/en_US/i/btn/btn_donateCC_LG.gif', alt:'Donate with PayPal button', title:'PayPal - The safer, easier way to pay online!', } }).render('#donate-button'); </script> </div> <a href="https://oeisf.org/donate/"> <strong>Other ways to Give</strong> </a> </div> </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> A016813 </div> <div class=seqname> a(n) = 4*n + 1. </div> </div> <div class=scorerefs> 247 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229, 233, 237</div> <div class=seqdatalinks> (<a href="/A016813/list">list</a>; <a href="/A016813/graph">graph</a>; <a href="/search?q=A016813+-id:A016813">refs</a>; <a href="/A016813/listen">listen</a>; <a href="/history?seq=A016813">history</a>; <a href="/search?q=id:A016813&fmt=text">text</a>; <a href="/A016813/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>Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 23 ).</div> <div class=sectline>Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 64 ).</div> <div class=sectline>Numbers k such that k and (k+1) have the same binary digital sum. - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Jun 05 2002</div> <div class=sectline>Numbers k such that (1 + sqrt(k))/2 is an algebraic integer. - <a href="/wiki/User:Alonso_del_Arte">Alonso del Arte</a>, Jun 04 2012</div> <div class=sectline>Numbers k such that 2 is the only prime p that satisfies the relationship p XOR k = p + k. - <a href="/wiki/User:Brad_Clardy">Brad Clardy</a>, Jul 22 2012</div> <div class=sectline>This may also be interpreted as the array T(n,k) = <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>(n+k) + <a href="/A008586" title="Multiples of 4.">A008586</a>(k) read by antidiagonals:</div> <div class=sectline> 1, 9, 21, 37, 57, 81, ...</div> <div class=sectline> 5, 17, 33, 53, 77, 105, ...</div> <div class=sectline> 13, 29, 49, 73, 101, 133, ...</div> <div class=sectline> 25, 45, 69, 97, 129, 165, ...</div> <div class=sectline> 41, 65, 93, 125, 161, 201, ...</div> <div class=sectline> 61, 89, 121, 157, 197, 241, ...</div> <div class=sectline> ...</div> <div class=sectline>- <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 10 2013</div> <div class=sectline>With leading term 2 instead of 1, 1/a(n) is the largest tolerance of form 1/k, where k is a positive integer, so that the nearest integer to (n - 1/k)^2 and to (n + 1/k)^2 is n^2. In other words, if interval arithmetic is used to square [n - 1/k, n + 1/k], every value in the resulting interval of length 4n/k rounds to n^2 if and only if k >= a(n). - <a href="/wiki/User:Rick_L._Shepherd">Rick L. Shepherd</a>, Jan 20 2014</div> <div class=sectline>Odd numbers for which the number of prime factors congruent to 3 (mod 4) is even. - <a href="/wiki/User:Daniel_Forgues">Daniel Forgues</a>, Sep 20 2014</div> <div class=sectline>For the Collatz conjecture, we identify two types of odd numbers. This sequence contains all the descenders: where (3*a(n) + 1) / 2 is even and requires additional divisions by 2. See <a href="/A004767" title="a(n) = 4*n + 3.">A004767</a> for the ascenders. - <a href="/wiki/User:Fred_Daniel_Kline">Fred Daniel Kline</a>, Nov 29 2014 [corrected by <a href="/wiki/User:Jaroslav_Krizek">Jaroslav Krizek</a>, Jul 29 2016]</div> <div class=sectline>a(n-1), n >= 1, is also the complex dimension of the manifold M(S), the set of all conjugacy classes of irreducible representations of the fundamental group pi_1(X,x_0) of rank 2, where S = {a_1, ..., a_{n}, a_{n+1} = oo}, a subset of P^1 = C U {oo}, X = X(S) = P^1 \ S, and x_0 a base point in X. See the Iwasaki et al. reference, Proposition 2.1.4. p. 150. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Apr 22 2016</div> <div class=sectline>For n > 3, also the number of (not necessarily maximal) cliques in the n-sunlet graph. - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Nov 29 2017</div> <div class=sectline>For integers k with absolute value in <a href="/A047202" title="Numbers that are congruent to {2, 3, 4} mod 5.">A047202</a>, also exponents of the powers of k having the same unit digit of k in base 10. - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Feb 23 2021</div> <div class=sectline>Starting with a(1) = 5, numbers ending with 01 in base 2. - <a href="/wiki/User:John_Keith">John Keith</a>, May 09 2022</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 150.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A016813/b016813.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>Colin Defant and Noah Kravitz, <a href="https://arxiv.org/abs/2201.03461">Loops and Regions in Hitomezashi Patterns</a>, arXiv:2201.03461 [math.CO], 2022. Theorem 1.3.</div> <div class=sectline>L. B. W. Jolley, <a href="https://www.scribd.com/document/22916829/Summation-of-Series-2nd-rev-ed-L-B-W-Jolley-1961-WW">Summation of Series</a>, Dover Publications, 1961, p. 16.</div> <div class=sectline>Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a></div> <div class=sectline>Konrad Knopp, <a href="http://name.umdl.umich.edu/ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")</div> <div class=sectline>Konrad Knopp, <a href="https://archive.org/details/theoryandapplica031692mbp/mode/2up">Theory and Application of Infinite Series</a>, Dover, p. 269.</div> <div class=sectline>Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.</div> <div class=sectline>William A. Stein's The Modular Forms Database, <a href="http://wstein.org/Tables/dimensions.html">PARI-readable dimension tables for Gamma_0(N)</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertNumber.html">Hilbert Number</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a></div> <div class=sectline>Wikipedia, <a href="http://en.wikipedia.org/wiki/Interval_arithmetic">Interval arithmetic</a></div> <div class=sectline><a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>(2*n).</div> <div class=sectline>Sum_{n>=0} (-1)^n/a(n) = (1/(4*sqrt(2)))*(Pi+2*log(sqrt(2)+1)) = <a href="/A181048" title="Decimal expansion of (log(1+sqrt(2))+Pi/2)/(2*sqrt(2)) = Sum_{k>=0} (-1)^k/(4*k+1).">A181048</a> [Jolley]. - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Apr 05 2002 [corrected by <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jul 30 2023]</div> <div class=sectline>G.f.: (1+3*x)/(1-x)^2. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Feb 27 2003 [corrected for offset 0 by <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 03 2014]</div> <div class=sectline>(1 + 5*x + 9*x^2 + 13*x^3 + ...) = (1 + 2*x + 3*x^2 + ...) / (1 - 3*x + 9*x^2 - 27*x^3 + ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jul 03 2003</div> <div class=sectline>a(n) = <a href="/A001969" title="Evil numbers: nonnegative integers with an even number of 1's in their binary expansion.">A001969</a>(n) + <a href="/A000069" title="Odious numbers: numbers with an odd number of 1's in their binary expansion.">A000069</a>(n). - <a href="/wiki/User:Philippe_Deléham">Philippe Deléham</a>, Feb 04 2004</div> <div class=sectline>a(n) = <a href="/A004766" title="Numbers whose binary expansion ends 01.">A004766</a>(n-1). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Oct 26 2008</div> <div class=sectline>a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=5. a(n) = 4 + a(n-1). - <a href="/wiki/User:Philippe_Deléham">Philippe Deléham</a>, Nov 03 2008</div> <div class=sectline><a href="/A056753" title="Only odd numbers occur and for all k there are k numbers between any two successive occurrences of k.">A056753</a>(a(n)) = 3. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Aug 23 2009</div> <div class=sectline><a href="/A179821" title="In binary representation of n: replace all blocks of k contiguous ones with binary representation of k.">A179821</a>(a(n)) = a(<a href="/A179821" title="In binary representation of n: replace all blocks of k contiguous ones with binary representation of k.">A179821</a>(n)). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 31 2010</div> <div class=sectline>a(n) = 8*n - 2 - a(n-1) for n > 0, a(0) = 1. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 20 2010</div> <div class=sectline>The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as a(n)^2 - <a href="/A002943" title="a(n) = 2*n*(2*n+1).">A002943</a>(n)*2^2 = 1. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Mar 11 2009 - Nov 25 2012</div> <div class=sectline><a href="/A089911" title="a(n) = Fibonacci(n) mod 12.">A089911</a>(6*a(n)) = 8. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 05 2013</div> <div class=sectline>a(n) = <a href="/A004767" title="a(n) = 4*n + 3.">A004767</a>(n) - 2. - <a href="/wiki/User:Jean-Bernard_François">Jean-Bernard François</a>, Sep 27 2013</div> <div class=sectline>a(n) = <a href="/A058281" title="Continued fraction for square root of e.">A058281</a>(3n+1). - <a href="/wiki/User:Eli_Jaffe">Eli Jaffe</a>, Jun 07 2016</div> <div class=sectline>From <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Jul 29 2016: (Start)</div> <div class=sectline>E.g.f.: (1 + 4*x)*exp(x).</div> <div class=sectline>a(n) = Sum_{k = 0..n} <a href="/A123932" title="a(0) = 1, a(n) = 4 for n > 0.">A123932</a>(k).</div> <div class=sectline>a(<a href="/A005098" title="Numbers k such that 4k + 1 is prime.">A005098</a>(k)) = x^2 + y^2.</div> <div class=sectline>Inverse binomial transform of <a href="/A014480" title="Expansion of (1+2*x)/(1-2*x)^2.">A014480</a>. (End)</div> <div class=sectline>Dirichlet g.f.: 4*Zeta(-1 + s) + Zeta(s). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Nov 02 2018</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Leo_Tavares">Leo Tavares</a>, Jul 02 2021: (Start)</div> <div class=sectline>Illustration of initial terms:</div> <div class=sectline> o</div> <div class=sectline> o o</div> <div class=sectline> o o o</div> <div class=sectline> o o o o o o o o o o o o o o o o</div> <div class=sectline> o o o</div> <div class=sectline> o o</div> <div class=sectline> o</div> <div class=sectline>(End)</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>seq(4*k+1, k=0..100); # <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Sep 28 2013</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Range[1, 237, 4] (* <a href="/wiki/User:Vladimir_Joseph_Stephan_Orlovsky">Vladimir Joseph Stephan Orlovsky</a>, May 26 2011 *)</div> <div class=sectline>Table[4 n + 1, {n, 0, 20}] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Nov 29 2017 *)</div> <div class=sectline>4 Range[0, 20] + 1 (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Nov 29 2017 *)</div> <div class=sectline>LinearRecurrence[{2, -1}, {5, 9}, {0, 20}] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Nov 29 2017 *)</div> <div class=sectline>CoefficientList[Series[(1 + 3 x)/(-1 + x)^2, {x, 0, 20}], x] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Nov 29 2017 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) [n: n in [1..250 by 4]];</div> <div class=sectline>(Haskell)</div> <div class=sectline>a016813 = (+ 1) . (* 4)</div> <div class=sectline>a016813_list = [1, 5 ..] -- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 14 2012</div> <div class=sectline>(PARI) a(n)=4*n+1 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Mar 22 2013</div> <div class=sectline>(PARI) x='x+O('x^100); Vec((1+3*x)/(1-x)^2) \\ <a href="/wiki/User:Altug_Alkan">Altug Alkan</a>, Oct 22 2015</div> <div class=sectline>(Scala) (0 to 59).map(4 * _ + 1) // <a href="/wiki/User:Alonso_del_Arte">Alonso del Arte</a>, Aug 08 2018</div> <div class=sectline>(GAP) List([0..70], n->4*n+1); # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Aug 08 2018</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Subsequence of <a href="/A042963" title="Numbers congruent to 1 or 2 mod 4.">A042963</a> and of <a href="/A079523" title="Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.">A079523</a>.</div> <div class=sectline>a(n) = <a href="/A093561" title="(4,1) Pascal triangle.">A093561</a>(n+1, 1), (4, 1)-Pascal column.</div> <div class=sectline>Cf. <a href="/A016921" title="a(n) = 6*n + 1.">A016921</a>, <a href="/A017281" title="a(n) = 10*n + 1.">A017281</a>, <a href="/A017533" title="a(n) = 12*n + 1.">A017533</a>, <a href="/A047202" title="Numbers that are congruent to {2, 3, 4} mod 5.">A047202</a>, <a href="/A158057" title="First differences of A051870: 16*n + 1.">A158057</a>, <a href="/A161705" title="a(n) = 18*n + 1.">A161705</a>, <a href="/A161709" title="a(n) = 22*n + 1.">A161709</a>, <a href="/A161714" title="a(n) = 28*n + 1.">A161714</a>, <a href="/A128470" title="a(n) = 30*n + 1.">A128470</a>. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jun 17 2009</div> <div class=sectline>Cf. <a href="/A004772" title="Numbers that are not congruent to 1 (mod 4).">A004772</a> (complement).</div> <div class=sectline>Cf. <a href="/A017557" title="a(n) = 12*n + 3.">A017557</a>.</div> <div class=sectline>Sequence in context: <a href="/A194395" title="Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=sqrt(14) and < > denotes fractional part.">A194395</a> <a href="/A162502" title="Inverse permutation to A162501.">A162502</a> <a href="/A004766" title="Numbers whose binary expansion ends 01.">A004766</a> * <a href="/A314668" title="Coordination sequence Gal.6.115.6 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number ...">A314668</a> <a href="/A314669" title="Coordination sequence Gal.6.216.6 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number ...">A314669</a> <a href="/A334526" title="Number of triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + 11xy = 0.">A334526</a></div> <div class=sectline>Adjacent sequences: <a href="/A016810" title="a(n) = (4n)^10.">A016810</a> <a href="/A016811" title="(4n)^11.">A016811</a> <a href="/A016812" title="(4n)^12.">A016812</a> * <a href="/A016814" title="a(n) = (4*n + 1)^2.">A016814</a> <a href="/A016815" title="a(n) = (4*n + 1)^3.">A016815</a> <a href="/A016816" title="a(n) = (4n+1)^4.">A016816</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></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>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 December 2 21:25 EST 2024. 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