A376785 Expansion of (1 + x - x^3)/((1 + x - x^3)^2 - 4*x).
1, 3, 5, 8, 19, 46, 98, 201, 429, 937, 2024, 4325, 9260, 19916, 42841, 91999, 197485, 424160, 911255, 1957402, 4203998, 9029425, 19394681, 41658577, 89478064, 192188361, 412801176, 886657848, 1904452689, 4090568027, 8786123349, 18871711384, 40534539675, 87064092870
Offset: 0
Keywords
Links
- Index entries for linear recurrences with constant coefficients, signature (2,-1,2,2,0,-1).
Programs
-
Mathematica
CoefficientList[Series[(1+x-x^3)/((1+x-x^3)^2-4x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2,2,0,-1},{1,3,5,8,19,46},40] (* Harvey P. Dale, Jun 29 2025 *) -
PARI
my(N=40, x='x+O('x^N)); Vec((1+x-x^3)/((1+x-x^3)^2-4*x)) -
PARI
a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, 2*k+1));
Formula
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,2*k+1).