OFFSET
1,2
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
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 linear recurrences with constant coefficients, signature (3,-3,1).
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
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)
E.g.f.: exp(x)*(1 + 7*x^2) - 1. - Stefano Spezia, Aug 01 2024
EXAMPLE
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)
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 *)
PROG
(PARI) a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Terrel Trotter, Jr., Apr 07 2002
STATUS
approved