A129362 a(n) = Sum_{k=floor((n+1)/2)..n} J(k+1), J(k) = A001045(k).

1, 1, 4, 8, 19, 37, 80, 160, 331, 661, 1344, 2688, 5419, 10837, 21760, 43520, 87211, 174421, 349184, 698368, 1397419, 2794837, 5591040, 11182080, 22366891, 44733781, 89473024, 178946048, 357903019, 715806037

Offset: 0

Author

Paul Barry, Apr 11 2007

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).

Crossrefs

Cf. A001045, A129361.

Programs

Magma

A001045:= func< n | (2^n - (-1)^n)/3 >;
[(&+[A001045(n-j+1): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jan 31 2024

Mathematica

LinearRecurrence[{1,3,-1,0,-2,-4},{1,1,4,8,19,37},30] (* Harvey P. Dale, Oct 22 2011 *)

SageMath

def A001045(n): return (2^n - (-1)^n)/3
def A129362(n): return sum(A001045(n-j+1) for j in range(1+(n//2)))
[A129362(n) for n in range(31)] # G. C. Greubel, Jan 31 2024

Formula

G.f.: (1+2*x^3)/((1-x-2*x^2)*(1-x^2-2*x^4)).

a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6).

a(n) = Sum_{k=0..n} ( J(k+1) - J((k+1)/2)*(1-(-1)^k)/2 ).

a(n) = Sum_{j=0..floor(n/2)} A001045(n-j+1). - G. C. Greubel, Jan 31 2024