A375317 Expansion of (1 + x)^2/(1 - x^2*(1 + x)^3).

1, 2, 2, 5, 11, 18, 34, 68, 126, 235, 450, 851, 1601, 3032, 5739, 10838, 20489, 38752, 73252, 138472, 261813, 494973, 935737, 1769080, 3344567, 6323023, 11953991, 22599701, 42725841, 80775310, 152709940, 288705927, 545813094, 1031887518, 1950836005, 3688154521

Offset: 0

Author

Seiichi Manyama, Aug 12 2024

Keywords

• nonn

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,3,3,1).

Crossrefs

Cf. A116090, A375315.

Programs

Mathematica

CoefficientList[Series[(1+x)^2/(1-x^2(1+x)^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,3,3,1},{1,2,2,5,11},40] (* Harvey P. Dale, Mar 25 2025 *)

PARI

my(N=40, x='x+O('x^N)); Vec((1+x)^2/(1-x^2*(1+x)^3))

PARI

a(n) = sum(k=0, n\2, binomial(3*k+2, n-2*k));

Formula

a(n) = a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).

a(n) = Sum_{k=0..floor(n/2)} binomial(3*k+2,n-2*k).

a(n) = A375315(n) + A375315(n-1).