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A154105 a(n) = 12*n^2 + 18*n + 7. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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

EXAMPLE

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.

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

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

Cf. A003215, A000290, A000217.

Sequence in context: A031395 A138906 A107938 * A159491 A106064 A282001

Adjacent sequences:  A154102 A154103 A154104 * A154106 A154107 A154108

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Jan 04 2009

STATUS

approved

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Last modified September 27 19:37 EDT 2022. Contains 357063 sequences. (Running on oeis4.)