A024458 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (Fibonacci numbers).

1, 1, 3, 5, 12, 19, 40, 65, 130, 210, 404, 654, 1227, 1985, 3653, 5911, 10720, 17345, 31090, 50305, 89316, 144516, 254568, 411900, 720757, 1166209, 2029095, 3283145, 5684340, 9197455, 15855964, 25655489, 44061862, 71293590, 122032508

Offset

1,3

Comments

From Wolfdieter Lang, Jan 02 2012: (Start)

chat(n):=a(n+1), n>=0, is the half-convolution of the sequence A000045(n+1), n>=0, with itself. For the definition of half-convolution see a comment on A201204, where also the rule to find the o.g.f. is given. Here the o.g.f. is obtained from (U(x)^2 + U2(x^2))/2 with U(x)=1/(1-x-x^2), the o.g.f. of A000045(n+1), n>=0, and U2(x):=(1-x)/((1+x)*(1-3*x+x^2)) the o.g.f. of A007598(n+1), n>=0. This coincides with the o.g.f. given below in the formula section after x has been divided.

For the bisection of this half-convolution see A027991(n+1) and A001870(n), n>=0.

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,0,-2,-3,1,1).

Formula

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

a(n) = ((13 - 5*(-1)^n + 10*n)*A000045(n) + (1 - (-1)^n + 2*n)*A000032(n) + 8*sin(Pi*n/2))/40. - Vladimir Reshetnikov, Oct 03 2016

From G. C. Greubel, Apr 06 2022: (Start)

a(2*n) = (1/5)*(n*Lucas(2*n+1) + Fibonacci(2*n)), n >= 1.

a(2*n+1) = (1/5)*((-1)^n + (n+1)*Lucas(2*n+2) + Fibonacci(2*n+1)), n >= 0.

a(n) = Sum_{j=0..floor((n-1)/2)} fibonacci(j+1)*Fibonacci(n-j). (End)

Mathematica

Table[((13-5(-1)^n +10n)Fibonacci[n] + (1-(-1)^n +2n)LucasL[n] +8Sin[Pi*n/2])/40, {n, 30}] (* Vladimir Reshetnikov, Oct 03 2016 *)
LinearRecurrence[{1, 3, -2, 0, -2, -3, 1, 1}, {1, 1, 3, 5, 12, 19, 40, 65}, 40] (* Harvey P. Dale, Mar 02 2023 *)

Prog

(Magma) [(&+[Fibonacci(j+1)*Fibonacci(n-j): j in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Apr 06 2022
(SageMath)
def A024458(n): return sum(fibonacci(j+1)*fibonacci(n-j) for j in (0..((n-1)//2)) )
[A024458(n) for n in (1..50)] # G. C. Greubel, Apr 06 2022

Crossrefs

Cf. A000032, A000045, A058071.

Sequence in context: A082740 A010067 A341710 * A143643 A321679 A266819

Adjacent sequences: A024455 A024456 A024457 * A024459 A024460 A024461

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from James A. Sellers, May 03 2000

Status

approved