A069127 Centered 14-gonal numbers.

1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, 6945, 7393, 7855, 8331, 8821, 9325, 9843, 10375, 10921, 11481, 12055, 12643, 13245, 13861, 14491

Offset: 1

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

• nonn
• easy
• nice

Comments

Binomial transform of [1, 14, 14, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 14, 0, 0, 0, ...]. - Gary W. Adamson, Jul 29 2011

Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - Omar E. Pol, Oct 03 2011

Examples

			a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
From _Bruno Berselli_, Oct 27 2017: (Start)
1   =         -(1) + (2).
15  =       -(1+2) + (3+4+5+6).
43  =     -(1+2+3) + (4+5+6+7+8+9+10).
85  =   -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Leo Tavares, Illustration: Heptagonal Stars.
• Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
• R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
• Index entries for sequences related to centered polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005448, A001844, A005891, A003215, A069099.

Cf. A163756, A193053.

Programs

Mathematica

FoldList[#1 + #2 &, 1, 14 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[14*Range[0,50]]+1 (* Harvey P. Dale, Apr 09 2012 *)

PARI

a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 7*n^2 - 7*n + 1.

a(n) = 14*n+a(n-1)-14 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

G.f.: -x*(1+12*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011

a(n) = A163756(n-1) + 1. - Omar E. Pol, Oct 03 2011

a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - Bruno Berselli, Oct 21 2011

Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - Vaclav Kotesovec, Jul 23 2019

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} a(n)/n! = 8*e - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)

a(n) = A069099(n) + 7*A000217(n-1). - Leo Tavares, Jul 09 2021

E.g.f.: exp(x)*(1 + 7*x^2) - 1. - Stefano Spezia, Aug 01 2024