A354855 a(n) = floor(n*(2+sqrt(5))^n), equivalently, floor(n*phi^(3n)), where phi = (1+sqrt(5))/2 is the golden ratio.
0, 4, 35, 228, 1287, 6820, 34667, 171332, 829455, 3952836, 18604979, 86693156, 400623383, 1838490212, 8387044091, 38065809540, 171999313951, 774138335108, 3472202765123, 15525625108324, 69229056160039, 307921937307684, 1366491508589195, 6051666872017348
Offset: 0
Links
- Stefano Spezia, Table of n, a(n) for n = 0..1500
- Index entries for linear recurrences with constant coefficients, signature (8,-13,-16,13,8,1).
Programs
-
Mathematica
a[n_] := Floor[n * GoldenRatio^(3*n)]; Array[a, 25, 0] (* Amiram Eldar, Jun 09 2022 *)
Formula
a(n) = floor((2+sqrt(5))^n*n).
a(n) = floor(n*phi^(3n)) where phi=(1+sqrt(5))/2 is the golden ratio.
a(n) = floor(n*F(3n-1)+n*phi*F(3n)), where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = n*L(3n) when n is odd and a(n) = n*L(3n)-1 when n is even (n>=2), where L(n) = A000032(n) is the n-th Lucas number.
G.f.: x*(4 + 3*x - 18*x^3 - 4*x^4 - x^5)/((1 - x)*(1 + x)*(1 - 4*x - x^2)^2). - Stefano Spezia, Jun 12 2022