OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1591
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 (8,-14,-8,-1).
FORMULA
a(n) = 8*a(n-1) - 14*a(n-2) - 8*a(n-3) - a(n-4) for n>=4, a(0)=a(1)=0, a(2)=1, a(3)=8.
a(n) = (4*(n-1)*a(n-1)+n*a(n-2))/(n-2) for n>=3, a(0)=a(1)=0, a(2)=1.
G.f.: z^2/(1-4*z-z^2)^2.
a(n) = ((n-1)*Fibonacci(3*n+3) + (n+1)*Fibonacci(3*n-3))/40.
Binet formula: a(n) = (1/(q^{3*(n+2)}*(q^3+q^{-3})^3))*(n*((-1)^{3*n+1}+q^{6*(n+2)})+(n+2)*((-1)^{3*n+1}q^6+q^{6*(n+1)})), where q=(1+sqrt(5))/2 is the golden ratio A001622.
Gegenbauer polynomials: a(n) = (-i)^{n-1} * C_{n-1}^{(2)}(2i).
E.g.f.: exp(2*x)*(10*x*cosh(sqrt(5)*x) + sqrt(5)*(5*x - 2)*sinh(sqrt(5)*x))/50. - Stefano Spezia, Apr 26 2026
MAPLE
a:= proc(n) option remember; `if`(n<4, [0$2, 1, 8][n+1],
8*a(n-1)-14*a(n-2)-8*a(n-3)-a(n-4))
end:
seq(a(n), n=0..25);
# Alternative:
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-8|-14|8>>^n)[2, 4]:
seq(a(n), n=0..25); # Alois P. Heinz, Apr 27 2026
MATHEMATICA
a[n_]:=((n-1)*Fibonacci[3*n+3]+(n+1)*Fibonacci[3*(n-1)])/40; Array[a, 26, 0] (* Stefano Spezia, Apr 26 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alex Stokolos, Apr 22 2026
STATUS
approved