A259546 a(n) = n^3*Fibonacci(n).
0, 1, 8, 54, 192, 625, 1728, 4459, 10752, 24786, 55000, 118459, 248832, 511901, 1034488, 2058750, 4042752, 7846061, 15069888, 28677479, 54120000, 101370906, 188586728, 348669719, 640991232, 1172265625, 2133603368, 3866095494, 6976587072, 12541531081
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 2.
- Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).
Programs
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Maple
a:= n-> n^3*(<<1|1>, <1|0>>^n)[1, 2]: seq(a(n), n=0..50); # Alois P. Heinz, Jun 30 2015
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Mathematica
Array[#^3*Fibonacci[#] &, 50, 0] (* Paolo Xausa, Jul 15 2024 *)
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PARI
a(n) = n^3*fibonacci(n)
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PARI
concat(0, Vec(x*(x^2+1)*(x^4-4*x^3+23*x^2+4*x+1)/(x^2+x-1)^4 + O(x^50)))
Formula
G.f.: x*(x^2+1)*(x^4-4*x^3+23*x^2+4*x+1) / (x^2+x-1)^4.
Sum_{k=1..n} a(k) = (n^3-6*n^2+24*n-50)*A000045(n+1) + ((n+1)^3-6*(n+1)^2+24*(n+1)-50)*A000045(n) + 50. - Prabha Sivaramannair, Jul 15 2024
E.g.f.: exp(x/2)*x*(5*(1 + x)*(1 + 2*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(1 + x*(9 + 4*x))*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Aug 25 2024