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.
(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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, May 03 2000
STATUS
approved