OFFSET
0,2
LINKS
Sergio Falcon, Sum of the Squares of the Extended (k, t)-Fibonacci Numbers, Preprints 2024, 2024081150. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,10,-2,-3,1).
FORMULA
a(n) = 4*(Fibonacci(n+1) - 1)*(Fibonacci(n+2) - 1) + n + 1 (see Falcon).
G.f.: (1 - 3*x + 7*x^2 - 3*x^3 + x^4 - x^5)/((1 - x)^2*(1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).
E.g.f.: (4*exp(-x) + exp(x)*(5 + x) + 8*exp(x/2)*((2*exp(x) - 5)*cosh(sqrt(5)*x/2) + sqrt(5)*(exp(x) - 2)*sinh(sqrt(5)*x/2)))/5.
a(n) = 5*a(n-1) - 6*a(n-2) - 4*a(n-3) + 10*a(n-4) - 2*a(n-5) - 3*a(n-6) + a(n-7). - Wesley Ivan Hurt, Feb 13 2026
MATHEMATICA
a[n_]:=4*(Fibonacci[n+1] - 1)*(Fibonacci[n+2] - 1) + n + 1; Array[a, 30, 0]
PROG
(Magma) [4*(Fibonacci(n+1) - 1)*(Fibonacci(n+2) - 1) + n + 1 : n in [0..40]]; // Wesley Ivan Hurt, Feb 13 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Aug 18 2024
STATUS
approved