A016278 Expansion of g.f. 1/((1-2x)(1-3x)(1-9x)).

1, 14, 145, 1370, 12541, 113534, 1023865, 9221090, 83008981, 747138854, 6724424785, 60520350410, 544684739821, 4902167424974, 44119521140905, 397075733249330, 3573681728253061, 32163135941435894, 289468224634660225

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
• Index entries for linear recurrences with constant coefficients, signature (14,-51,54).

Crossrefs

Cf. A053141, A053142.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-3*x)*(1-9*x)))); // Vincenzo Librandi, Jun 24 2013

Mathematica

CoefficientList[Series[1 / ((1 - 2 x) (1 - 3 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

PARI

Vec(1/((1-2*x)*(1-3*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 14*a(n-1) - 51*a(n-2) + 54*a(n-3); a(n) = (4/7)*2^(n-1) + (-3/2)*3^(n-1) + (27/14)*9^(n-1). - Antonio Alberto Olivares, Apr 21 2008, Apr 22 2008

E.g.f.: exp(2*x)*(8 - 21*exp(x) + 27*exp(7*x))/14. - Stefano Spezia, Feb 12 2025