cp's OEIS Frontend

A069129 Centered 16-gonal numbers.

%I A069129 #110 Feb 16 2025 08:32:45
%S A069129 1,17,49,97,161,241,337,449,577,721,881,1057,1249,1457,1681,1921,2177,
%T A069129 2449,2737,3041,3361,3697,4049,4417,4801,5201,5617,6049,6497,6961,
%U A069129 7441,7937,8449,8977,9521,10081,10657,11249,11857,12481,13121,13777,14449,15137,15841
%N A069129 Centered 16-gonal numbers.
%C A069129 Also, sequence found by reading the line from 1, in the direction 1, 17, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139098 in the same spiral. - _Omar E. Pol_, Apr 26 2008
%C A069129 The subsequence of primes begins: 17, 97, 241, 337, 449, 577, 881, 1249, 3041, 3361, 3697, 4049, 4801, 6961, 7937, 9521,  10657, 13121, 14449. See A184899: n such that the n-th centered 12-gonal number is prime. Indices of prime star numbers. - _Jonathan Vos Post_, Feb 27 2011
%C A069129 Binomial transform of [1, 16, 16, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 16, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 28 2011
%C A069129 Centered hexadecagonal numbers or centered hexakaidecagonal numbers. - _Omar E. Pol_, Oct 03 2011
%C A069129 a(n) = m(n,n) for an array constructed by using the terms in A016813 as the antidiagonals; the first few antidiagonals are 1; 5,9; 13,17,21; 25,29,33,37. - _J. M. Bergot_, Jul 05 2013
%C A069129 [The first five rows begin: 1,9,21,37,57; 5,17,33,53,77; 13,29,49,73,101; 25,45,69,97,129; 41,65,93,125,161.]
%H A069129 Vincenzo Librandi, <a href="/A069129/b069129.txt">Table of n, a(n) for n = 1..1000</a>
%H A069129 Omar E. Pol, <a href="http://polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A069129 Leo Tavares, <a href="/A069129/a069129.jpg">Illustration: Octagonal Stars</a>.
%H A069129 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.
%H A069129 R. Yin, J. Mu, and T. Komatsu, <a href="https://doi.org/10.20944/preprints202407.2280.v1">The p-Frobenius Number for the Triple of the Generalized Star Numbers</a>, Preprints 2024, 2024072280. See p. 2.
%H A069129 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.
%H A069129 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A069129 a(n) = 8*n^2 - 8*n + 1.
%F A069129 a(n) = A035008(n-1) + 1. - _Omar E. Pol_, Apr 26 2008
%F A069129 a(n) = 16*n + a(n-1) - 16 with n > 1, a(1)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A069129 G.f.: -x*(1+14*x+x^2) / (x-1)^3. - _R. J. Mathar_, Feb 04 2011
%F A069129 E.g.f.: (8*x^2 + 1)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F A069129 a(n) = A056220(2n-1). - _Bruce J. Nicholson_, Aug 31 2017
%F A069129 Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(2))) / (4*sqrt(2)). - _Vaclav Kotesovec_, Jul 23 2019
%F A069129 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A069129 Sum_{n>=1} a(n)/n! = 9*e - 1.
%F A069129 Sum_{n>=1} (-1)^n * a(n)/n! = 9/e - 1. (End)
%F A069129 Product_{n>=2} (a(n) - 1) / (a(n) + 1) = Pi/4. - _Dimitris Valianatos_, Jun 27 2020
%F A069129 a(n) = A016754(n-1) + 8*A000217(n-1). - _Leo Tavares_, Jul 19 2021
%e A069129 a(5) = 161 because 8*5^2 - 8*5 + 1 = 200 - 40 + 1 = 161.
%t A069129 FoldList[#1 + #2 &, 1, 16 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A069129 Rest[CoefficientList[Series[-x(1+14x+x^2)/(x-1)^3,{x,0,50}],x]]  (* _Harvey P. Dale_, Apr 22 2011 *)
%o A069129 (Magma) [8*n^2-8*n+1: n in [0..50]]; // _Vincenzo Librandi_, Feb 05 2013
%o A069129 (PARI) a(n)=8*n^2-8*n+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A069129 Cf. A005448, A001844, A005891, A003215, A069099, A000217, A035008, A139098.
%Y A069129 Bisection of A077221.
%K A069129 easy,nice,nonn
%O A069129 1,2
%A A069129 _Terrel Trotter, Jr._, Apr 07 2002