A120756 Expansion of g.f. x*(1 +82*x +366*x^2 -189*x^3)/((1+3*x)*(1+9*x)*(1-21*x)).

1, 91, 1410, 33543, 670734, 14383251, 299383290, 6310987263, 132315428934, 2780561320011, 58374353168370, 1226018323723383, 25744972644839934, 540657134887786371, 11353685448630220650, 238428423877048161903, 5006987636316607975734

Offset

1,2

Comments

Former name: a(n) = leftmost term of M^n * [1,0,0,0,0,0] where M is the 6 X 6 matrix [1,2,3,4,5,6; 2,3,1,5,6,4; 3,1,2,6,4,5; 4,6,5,1,3,2; 5,4,6,2,1,3; 6,5,4,3,2,1].

M is a multiplication table for the symmetric group of degree 3 (S_3).

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (9,225,567).

Formula

G.f.: x*(1 +82*x +366*x^2 -189*x^3)/((1+3*x)*(1+9*x)*(1-21*x)). - Colin Barker, Dec 13 2012

From G. C. Greubel, Nov 14 2022: (Start)

a(n) = (9*(21)^n + 9*(-9)^n + 24*(-3)^n + 18*[n=1] - 42*[n=0])/54.

E.g.f.: (1/54)*(-42 + 18*x + 24*exp(-3*x) + 9*exp(-9*x) + 9*exp(21*x)). (End)

Mathematica

LinearRecurrence[{9, 225, 567}, {1, 91, 1410, 33543}, 20] (* Harvey P. Dale, Apr 11 2018 *)
M= {{1, 2, 3, 4, 5, 6}, {2, 3, 1, 5, 6, 4}, {3, 1, 2, 6, 4, 5}, {4, 6, 5, 1, 3, 2}, {5, 4, 6, 2, 1, 3}, {6, 5, 4, 3, 2, 1}};
A120756[n_]:= A120756[n]= MatrixPower[M, n][[1, 1]];
Table[A120756[n], {n, 30}] (* G. C. Greubel, Nov 14 2022 *)

Prog

(Magma) I:=[91, 1410, 33543]; [1] cat [n le 3 select I[n] else 9*Self(n-1) + 225*Self(n-2) +567*Self(n-3): n in [1..31]]; // G. C. Greubel, Nov 14 2022
(SageMath)
def A120756(n): return (9*(21)^n +9*(-9)^n +24*(-3)^n +18*int(n==1) -42*int(n==0))/54
[A120756(n) for n in range(1, 30)] # G. C. Greubel, Nov 14 2022

Crossrefs

Sequence in context: A221815 A195221 A213559 * A027787 A333112 A140667

Adjacent sequences: A120753 A120754 A120755 * A120757 A120758 A120759

Keyword

nonn,easy

Author

Gary W. Adamson and Roger L. Bagula, Jul 01 2006

Extensions

Edited by N. J. A. Sloane, May 06 2010

Name changed by G. C. Greubel, Nov 14 2022

Status

approved