A375500 a(n) = Sum_{k=0..n} A001595(k)^2.
1, 2, 11, 36, 117, 342, 967, 2648, 7137, 19018, 50347, 132716, 348941, 915950, 2401911, 6294640, 16489889, 43187778, 113094099, 296127940, 775343821, 2029991062, 5314771031, 13914551256, 36429253657, 95373809882, 249693147107, 653707202748, 1711431003597, 4480589921838
Offset: 0
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).
Programs
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Mathematica
a[n_]:=4*(Fibonacci[n+1] - 1)*(Fibonacci[n+2] - 1) + n + 1; Array[a,30,0]
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.