OFFSET
0,1
COMMENTS
a(n) is the number of partitions with three integral dissimilar components of the number 12(n+1), e.g for n=0, 12 may be partitioned in the 7 ways (1,2,9), (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5). - Ian Duff, Jan 31 2010
Sequence found by reading the line from 7, in the direction 7, 37, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, May 08 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..3000
John Elias, Animated Illustration: Starburst Hexagrams
Leo Tavares, Illustration: Hexagonal Halos
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (7 + 16*x + x^2)/(1-x)^3.
a(n) = 6*A014106(n) + 7.
a(0) = 7; for n > 0, a(n) = a(n-1) + 24*n + 6.
a(-n-1) = 2*A085473(n) - 1. - Bruno Berselli, Sep 05 2011
E.g.f.: (7 + 30*x + 12*x^2)*exp(x). - G. C. Greubel, Sep 02 2016
a(n) = 1 + A152746(n+1). - Omar E. Pol, May 08 2018
EXAMPLE
MATHEMATICA
Table[12*n^2 + 18*n + 7, {n, 0, 42}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
LinearRecurrence[{3, -3, 1}, {7, 37, 91}, 25] (* G. C. Greubel, Sep 02 2016 *)
PROG
(Magma) [ 12*n^2+18*n+7: n in [0..40] ];
(PARI) a(n)=12*n^2+18*n+7 \\ Charles R Greathouse IV, Sep 02 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Jan 04 2009
STATUS
approved