OFFSET
0,4
COMMENTS
Part of a family of self-convolutions of Fibonacci-related sequences F(k*n)/F(k).
Related to A395431, which is the self-convolution of F(3*n)/F(3).
LINKS
D. Dmytryshyn, D. Gray, V. Khamitov, and Alex Stokolos, Convolved numbers of k-section of the Fibonacci sequence: properties, consequences, arXiv:2603.08636 [math.CA], 2026; see also alternate link, Herald of Advanced Information Technology. 2026. Vol. 9, No. 2. P. 129-139.
Index entries for linear recurrences with constant coefficients, signature (14,-51,14,-1).
FORMULA
a(n) = 14*a(n-1) - 51*a(n-2) + 14*a(n-3) - a(n-4) for n >= 4.
a(n) = (7*(n-1)*a(n-1) - n*a(n-2)) / (n-2) for n >= 3, a(0)=a(1)=0, a(2)=1.
a(n) = Sum_{j=1..n} b(j)*b(n-j), where b(j) = F(4*j)/3.
a(n) = GegenbauerC(n-2, 2, 7/2) for n >= 2.
a(n) = (q^(-4*(n+1)) / (q^4 - q^(-4))^3) * ((n-1)*(q^(8*(n+1)) - 1) - (n+1)*(q^(8*n) - q^8)), where q = (1+sqrt(5))/2.
a(n) = ((n-1)*F(4*n+4) - (n+1)*F(4*n-4)) / 135.
G.f.: x^2 / (1 - 7*x + x^2)^2.
E.g.f.: exp(7*x/2) * ( (7*x/45)*cosh(3*sqrt(5)*x/2) + (sqrt(5)*x/15 - 14*sqrt(5)/675)*sinh(3*sqrt(5)*x/2) ).
MAPLE
N := 15: a[0] := 0: a[1] := 0: a[2] := 1: a[3] := 14:
for j from 4 to N do
a[j] := 14*a[j - 1] - 51*a[j - 2] + 14*a[j - 3] - a[j - 4]:
end do:
seq(a[k], k = 0 .. N);
MATHEMATICA
LinearRecurrence[{14, -51, 14, -1}, {0, 0, 1, 14}, 24] (* Amiram Eldar, May 17 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alex Stokolos, May 16 2026
STATUS
approved