A069129 Centered 16-gonal numbers.

1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, 7937, 8449, 8977, 9521, 10081, 10657, 11249, 11857, 12481, 13121, 13777, 14449

Offset

1,2

Comments

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

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

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

Centered hexadecagonal numbers or centered hexakaidecagonal numbers. - Omar E. Pol, Oct 03 2011

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

[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.]

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Leo Tavares, Illustration: Octagonal Stars

Eric Weisstein's World of Mathematics, Centered Polygonal Numbers

Index entries for sequences related to centered polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 8*n^2 - 8*n + 1.

a(n) = A035008(n-1) + 1. - Omar E. Pol, Apr 26 2008

a(n) = 16*n + a(n-1) - 16 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010

G.f.: -x*(1+14*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011

E.g.f.: (8*x^2 + 1)*exp(x). - G. C. Greubel, Jul 18 2017

a(n) = A056220(2n-1). - Bruce J. Nicholson, Aug 31 2017

Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(2))) / (4*sqrt(2)). - Vaclav Kotesovec, Jul 23 2019

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} a(n)/n! = 9*e - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 9/e - 1. (End)

Product_{n>=2} (a(n) - 1) / (a(n) + 1) = Pi/4. - Dimitris Valianatos, Jun 27 2020

a(n) = A016754(n-1) + 8*A000217(n-1). - Leo Tavares, Jul 19 2021

Example

a(5) = 161 because 8*5^2 - 8*5 + 1 = 200 - 40 + 1 = 161.

Mathematica

FoldList[#1 + #2 &, 1, 16 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Rest[CoefficientList[Series[-x(1+14x+x^2)/(x-1)^3, {x, 0, 50}], x]]  (* Harvey P. Dale, Apr 22 2011 *)

Prog

(Magma) [8*n^2-8*n+1: n in [0..50]]; // Vincenzo Librandi, Feb 05 2013
(PARI) a(n)=8*n^2-8*n+1 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A005448, A001844, A005891, A003215, A069099, A000217, A035008, A139098.

Bisection of A077221.

Sequence in context: A239130 A181426 A029487 * A176273 A124710 A113867

Adjacent sequences: A069126 A069127 A069128 * A069130 A069131 A069132

Keyword

easy,nice,nonn

Author

Terrel Trotter, Jr., Apr 07 2002

Status

approved