A271827 Expansion of (x^5-2*x^4+2*x^3-x+1)/(x^4-2*x^3+3*x^2-3*x+1).

1, 2, 3, 7, 13, 23, 41, 73, 129, 227, 399, 701, 1231, 2161, 3793, 6657, 11683, 20503, 35981, 63143, 110809, 194457, 341249, 598851, 1050911, 1844221, 3236383, 5679457, 9966753, 17490433, 30693571, 53863463, 94523789, 165877687, 291095049

Offset

0,2

Links

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-1).

Formula

a(n) = 1+2*Sum_{k=1..n} Sum_{j=0..(n-k)/2} binomial(k+1,j)*binomial(n-k-j-1,n-k-2*j), a(1)=2.

Mathematica

CoefficientList[Series[(x^5 - 2 x^4 + 2 x^3 - x + 1)/(x^4 - 2 x^3 + 3 x^2 - 3 x + 1), {x, 0, 34}], x] (* or *)
Table[If[n == 1, 2, 2 (Sum[Sum[Binomial[k + 1, j] Binomial[n - k - j - 1, n - k - 2 j], {j, 0, (n - k)/2}], {k, n}]) + 1], {n, 0, 34}] (* Michael De Vlieger, Apr 15 2016 *)
LinearRecurrence[{3, -3, 2, -1}, {1, 2, 3, 7, 13, 23}, 40] (* Harvey P. Dale, Oct 20 2023 *)

Prog

(Maxima)
a(n):=if n=1 then 2 else 2*(sum(sum(binomial(k+1, j)*binomial(n-k-j-1, n-k-2*j), j, 0, (n-k)/2), k, 1, n))+1;
(PARI) x='x+O('x^99); Vec((x^5-2*x^4+2*x^3-x+1)/(x^4-2*x^3+3*x^2-3*x+1)) \\ Altug Alkan, Apr 15 2016

Crossrefs

Sequence in context: A208149 A144104 A088175 * A298339 A091440 A175211

Adjacent sequences: A271824 A271825 A271826 * A271828 A271829 A271830

Keyword

nonn,easy

Author

Vladimir Kruchinin, Apr 15 2016

Status

approved