# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a001106 Showing 1-1 of 1 %I A001106 M4604 #184 Feb 16 2025 08:32:22 %S A001106 0,1,9,24,46,75,111,154,204,261,325,396,474,559,651,750,856,969,1089, %T A001106 1216,1350,1491,1639,1794,1956,2125,2301,2484,2674,2871,3075,3286, %U A001106 3504,3729,3961,4200,4446,4699,4959,5226,5500,5781,6069,6364 %N A001106 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2. %C A001106 Sequence found by reading the line from 0, in the direction 0, 9, ... and the parallel line from 1, in the direction 1, 24, ..., in the square spiral whose vertices are the generalized 9-gonal (enneagonal) numbers A118277. Also sequence found by reading the same lines in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - _Omar E. Pol_, Sep 10 2011 %C A001106 Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and x+y <= n. - _Reinhard Zumkeller_, Jan 23 2012 %C A001106 Partial sums give A007584. - _Omar E. Pol_, Jan 15 2013 %D A001106 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. %D A001106 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6. %D A001106 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A001106 T. D. Noe and William A. Tedeschi, <a href="/A001106/b001106.txt">Table of n, a(n) for n = 0..10000</a> (1000 terms were computed by T. D. Noe) %H A001106 S. Barbero, U. Cerruti and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4. %H A001106 C. K. Cook and M. R. Bacon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343. %H A001106 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=343">Encyclopedia of Combinatorial Structures 343</a> %H A001106 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. %H A001106 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992 %H A001106 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalNumber.html">Nonagonal Number</a>. %H A001106 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a> %H A001106 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1). %F A001106 a(n) = (7*n - 5)*n/2. %F A001106 G.f.: x*(1+6*x)/(1-x)^3. - _Simon Plouffe_ in his 1992 dissertation. %F A001106 a(n) = n + 7*A000217(n-1). - _Floor van Lamoen_, Oct 14 2005 %F A001106 Starting (1, 9, 24, 46, 75, ...) gives the binomial transform of (1, 8, 7, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 22 2007 %F A001106 Row sums of triangle A131875 starting (1, 9, 24, 46, 75, 111, ...). A001106 = binomial transform of (1, 8, 7, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 22 2007 %F A001106 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 1, a(2) = 9. - _Jaume Oliver Lafont_, Dec 02 2008 %F A001106 a(n) = 2*a(n-1) - a(n-2) + 7. - _Mohamed Bouhamida_, May 05 2010 %F A001106 a(n) = a(n-1) + 7*n - 6 (with a(0) = 0). - _Vincenzo Librandi_, Nov 12 2010 %F A001106 a(n) = A174738(7n). - _Philippe Del茅ham_, Mar 26 2013 %F A001106 a(7*a(n) + 22*n + 1) = a(7*a(n) + 22*n) + a(7*n+1). - _Vladimir Shevelev_, Jan 24 2014 %F A001106 E.g.f.: x*(2 + 7*x)*exp(x)/2. - _Ilya Gutkovskiy_, Jul 28 2016 %F A001106 a(n+2) + A000217(n) = (2*n+3)^2. - _Ezhilarasu Velayutham_, Mar 18 2020 %F A001106 Product_{n>=2} (1 - 1/a(n)) = 7/9. - _Amiram Eldar_, Jan 21 2021 %F A001106 Sum_{n>=1} 1/a(n) = A244646. - _Amiram Eldar_, Nov 12 2021 %F A001106 a(n) = A000217(3*n-2) - (n-1)^2. - _Charlie Marion_, Feb 27 2022 %F A001106 a(n) = 3*A000217(n) + 2*A005563(n-2). In general, if P(k,n) = the n-th k-gonal number, then P(m*k,n) = m*P(k,n) + (m-1)*A005563(n-2). - _Charlie Marion_, Feb 21 2023 %t A001106 Table[n(7n - 5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* _Harvey P. Dale_, Nov 06 2011 *) %t A001106 (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[9], n], {n, 0, 43}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *) %t A001106 PolygonalNumber[9,Range[0,50]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Nov 19 2019 *) %o A001106 (PARI) a(n)=n*(7*n-5)/2 \\ _Charles R Greathouse IV_, Jun 10 2011 %o A001106 (Haskell) %o A001106 a001106 n = length [(x,y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n] %o A001106 -- _Reinhard Zumkeller_, Jan 23 2012 %o A001106 (Haskell) a001106 n = n*(7*n-5) `div` 2 -- _James Spahlinger_, Oct 18 2012 %o A001106 (Python) %o A001106 def aList(): # Intended to compute the initial segment of the sequence, not isolated terms. %o A001106 x, y = 1, 1 %o A001106 yield 0 %o A001106 while True: %o A001106 yield x %o A001106 x, y = x + y + 7, y + 7 %o A001106 A001106 = aList() %o A001106 print([next(A001106) for i in range(49)]) # _Peter Luschny_, Aug 04 2019 %Y A001106 Cf. A093564 ((7, 1) Pascal, column m=2). Partial sums of A016993. %Y A001106 Cf. A131875, A057655, A069099, A244646. %K A001106 nonn,easy,nice %O A001106 0,3 %A A001106 _N. J. A. Sloane_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE