A154105 a(n) = 12*n^2 + 18*n + 7.

7, 37, 91, 169, 271, 397, 547, 721, 919, 1141, 1387, 1657, 1951, 2269, 2611, 2977, 3367, 3781, 4219, 4681, 5167, 5677, 6211, 6769, 7351, 7957, 8587, 9241, 9919, 10621, 11347, 12097, 12871, 13669, 14491, 15337, 16207, 17101, 18019, 18961, 19927, 20917, 21931

Offset: 0

Author

Klaus Brockhaus, Jan 04 2009

Keywords

• nonn
• easy

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

Examples

			a(2) = 12*2^2 + 18*2 + 7 = 91 = 6*14 + 7 = 6*A014106(2) + 7.
a(3) = a(2) + 24*3 + 6 = 91 + 72 + 6 = 169.
a(-4) = 12*4^2 - 18*4 + 7 = 127 = 2*64 - 1 = 2*A085473(3) - 1.
		

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).

Crossrefs

Cf. A001082, A014106, A152746, A153286, A085473.

Cf. A003215, A000290, A000217.

Programs

Magma

[ 12*n^2+18*n+7: n in [0..40] ];

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 *)

PARI

a(n)=12*n^2+18*n+7 \\ Charles R Greathouse IV, Sep 02 2016

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

a(n) = A003215(n) + 6*A000290(n+1) + 6*A000217(n). - Leo Tavares, Sep 12 2022