A085281 Expansion of (1 - 3*x + x^2)/((1-2*x)*(1-3*x)).

1, 2, 5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697, 847322163875, 2541932937193, 7625731702715, 22877060890417, 68630914235795, 205892205836473

Offset: 0

Author

Paul Barry, Jun 25 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A005578.

Binomial transform is A085282.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (5,-6).

Crossrefs

Cf. A005578, A007689, A085282.

Programs

Magma

[3^n/3+2^n/2+0^n/6: n in [0..40]]; // Vincenzo Librandi, May 29 2011

Mathematica

a[n_]:=3^n/3 + 2^n/2; Flatten[Join[{1, Array[a, 50]}]] (* or *)
CoefficientList[Series[(1 - 3*x + x^2)/((1-2*x)*(1-3*x)), {x, 0, 50}], x] (* Stefano Spezia, Sep 09 2018 *)
LinearRecurrence[{5,-6},{1,2,5},40] (* Harvey P. Dale, Jun 14 2022 *)

SageMath

def A085281(n): return 2^(n-1) +3^(n-1) +int(n==0)/6
[A085281(n) for n in range(41)] # G. C. Greubel, Nov 11 2024

Formula

a(n) = 3^(n-1) + 2^(n-1) + 0^n/6.

a(n) = A007689(n-1), n > 0. - R. J. Mathar, Sep 12 2008

E.g.f.: (1/6)*(1 + 3*exp(2*x) + 2*exp(3*x)). - G. C. Greubel, Nov 11 2024