A136305 Expansion of g.f. (3 -x +2*x^2)/(1 -3*x +2*x^2 -x^3).

3, 8, 20, 47, 109, 253, 588, 1367, 3178, 7388, 17175, 39927, 92819, 215778, 501623, 1166132, 2710928, 6302143, 14650705, 34058757, 79177004, 184064203, 427897358, 994740672, 2312491503, 5375890523, 12497429235, 29052998162, 67540026539, 157011512528

Offset: 0

Author

Richard Choulet, Mar 22 2008

Keywords

• nonn
• easy

Comments

Previous name: Transform of A000027 by the T_{1,2} transformation (see link).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Richard Choulet, Curtz-like transformation.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Crossrefs

Cf. A097550, A135364, A136302, A136303, A136304, A137229, A137234, A137249.

Programs

Magma

[n le 3 select 2^(n-1)*(n+2) else 3*Self(n-1) - 2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 19 2021

Mathematica

LinearRecurrence[{3,-2,1}, {3,8,20}, 40] (* G. C. Greubel, Apr 19 2021 *)
CoefficientList[Series[(3-x+2x^2)/(1-3x+2x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Oct 15 2021 *)

Sage

@CachedFunction
def a(n): return 2^n*(n+3) if n<3 else sum((-1)^j*(3-j)*a(n-j-1) for j in (0..2))
[a(n) for n in (0..40)] # G. C. Greubel, Apr 19 2021

Formula

G.f.: f(z) = 3 +8*z + ... = (3 -z +2*z^2)/(1 -3*z +2*z^2 -z^3).

a(n+3) = 3*a(n+2) -2*a(n+1) +a(n) (n>=0). - Richard Choulet, Apr 07 2009