A068451 Factorial expansion of the golden ratio (1+sqrt(5))/2 = Sum_{n>=1} a(n)/n!.

1, 1, 0, 2, 4, 0, 6, 7, 1, 1, 8, 1, 6, 0, 11, 0, 10, 5, 6, 9, 15, 20, 10, 15, 1, 18, 5, 13, 9, 0, 13, 15, 2, 27, 28, 2, 32, 36, 11, 4, 34, 37, 0, 4, 32, 10, 4, 4, 32, 46, 39, 37, 2, 20, 27, 8, 54, 27, 45, 9, 26, 18, 59, 0, 22, 63, 41, 54, 65, 61, 45, 51, 61, 31, 68, 48, 34, 39, 71, 59

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index to sequences related to factorial base representation of noninteger constants.

Crossrefs

Cf. A001622 (decimal expansion).

Cf. A075874 and A007514.

Programs

Magma

SetDefaultRealField(RealField(250));  [Floor((1+Sqrt(5))/2)] cat [Floor(Factorial(n)*(1+Sqrt(5))/2) - n*Floor(Factorial((n-1))*(1+Sqrt(5))/2) : n in [2..80]]; // G. C. Greubel, Mar 21 2018

Mathematica

With[{b = GoldenRatio}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Mar 21 2018 *)

PARI

default(realprecision, 250); b = (1+sqrt(5))/2; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Mar 21 2018

PARI

A068451(N=90,c=precision(sqrt(5)+1,logint(N!,10))/2)=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ M. F. Hasler, Nov 27 2018

Sage

def A068451(n):
    if (n==1): return floor(golden_ratio)
    else: return expand(floor(factorial(n)*golden_ratio) - n*floor(factorial(n-1)*golden_ratio))
[A068451(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018