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a(n) = Sum_{k=0..n} A001595(k)^2.
1

%I #8 Aug 18 2024 20:29:25

%S 1,2,11,36,117,342,967,2648,7137,19018,50347,132716,348941,915950,

%T 2401911,6294640,16489889,43187778,113094099,296127940,775343821,

%U 2029991062,5314771031,13914551256,36429253657,95373809882,249693147107,653707202748,1711431003597,4480589921838

%N a(n) = Sum_{k=0..n} A001595(k)^2.

%H Sergio Falcon, <a href="https://doi.org/10.20944/preprints202408.1150.v1">Sum of the Squares of the Extended (k, t)-Fibonacci Numbers</a>, Preprints 2024, 2024081150. See p. 4.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-4,10,-2,-3,1).

%F a(n) = 4*(Fibonacci(n+1) - 1)*(Fibonacci(n+2) - 1) + n + 1 (see Falcon).

%F 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)).

%F 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.

%t a[n_]:=4*(Fibonacci[n+1] - 1)*(Fibonacci[n+2] - 1) + n + 1; Array[a,30,0]

%Y Cf. A000045, A001595, A375501.

%K nonn,easy

%O 0,2

%A _Stefano Spezia_, Aug 18 2024