A024551 a(n) = floor(a(n-1)/(sqrt(5) - 2)) for n > 0 and a(0) = 1.

1, 4, 16, 67, 283, 1198, 5074, 21493, 91045, 385672, 1633732, 6920599, 29316127, 124185106, 526056550, 2228411305, 9439701769, 39987218380, 169388575288, 717541519531, 3039554653411, 12875760133174, 54542595186106, 231046140877597

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Programs

Mathematica

a[0] = 1;
a[n_] := Floor[a[n - 1]/FractionalPart[Sqrt[5]]]
Table[a[n], {n, 0, 60}]
(* Clark Kimberling, Aug 16 2012 *)
a[0]=1;
a[1]=4;
a[2]=16;
a[n_]:=Floor[a[n-1]^2/a[n-2]]+3
Table[a[n],{n,0,60}]
With[{c=Sqrt[5]-2},NestList[Floor[#/c]&,1,30]] (* Harvey P. Dale, Jul 18 2018 *)

PARI

a(n)=([0,1,0; 0,0,1; -1,-3,5]^n*[1;4;16])[1,1] \\ Charles R Greathouse IV, Jan 20 2017

PARI

step(n)=2*n + sqrtint(5*n^2)
a(n)=if(n, step(a(n-1)), 1) \\ Charles R Greathouse IV, Jan 20 2017

Formula

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - Clark Kimberling, Aug 16 2012

G.f.: (-x^2-x+1)/[(1-x)(1-4x-x^2)].

a(n) = (3*Fibonacci(3*n+2) + 1)/4 = 1 + 3*Sum_{k=0..n} A001076(k). - Ehren Metcalfe, Apr 15 2019