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

0,2

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).

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

Mathematica

a[n_] := Floor[n * GoldenRatio^(3*n)]; Array[a, 25, 0] (* Amiram Eldar, Jun 09 2022 *)

Crossrefs

Cf. A000032, A000045, A001622, A004976, A098317, A128439.

Sequence in context: A362346 A005552 A127519 * A128811 A220256 A220320

Adjacent sequences: A354852 A354853 A354854 * A354856 A354857 A354858

Keyword

nonn,easy

Author

Jiale Wang, Jun 09 2022

Status

approved