A172128 a(n) = floor(phi^n/n), where phi = golden ratio = (1+sqrt(5))/2.

1, 1, 1, 1, 2, 2, 4, 5, 8, 12, 18, 26, 40, 60, 90, 137, 210, 320, 492, 756, 1165, 1800, 2786, 4320, 6710, 10440, 16266, 25380, 39650, 62016, 97108, 152213, 238824, 375060, 589521, 927368, 1459960, 2300100, 3626213, 5720653, 9030450, 14263680, 22542396, 35645400

Offset: 1

Author

Clark Kimberling, Nov 20 2010

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..4500

Crossrefs

Cf. A000045, A001622 (phi), A181885.

Programs

Magma

[Floor((Lucas(n) + Sqrt(5)*Fibonacci(n))/(2*n)): n in [1..50]]; // G. C. Greubel, Apr 17 2022

Mathematica

Table[Floor[((1 + Sqrt[5])/2)^n/n], {n, 1, 50}]
Table[Floor[GoldenRatio^n/n],{n,50}] (* Harvey P. Dale, Dec 12 2018 *)

SageMath

[floor(golden_ratio^n/n) for n in (1..50)] # G. C. Greubel, Apr 17 2022

Formula

a(n) = floor((1/n)*(Fibonacci(n)*phi + Fibonacci(n-1))), where phi = (1+sqrt(5))/2.