A322119 Factorial expansion of (1-sqrt(5))/2 = Sum_{n>=1} a(n)/n!.

-1, 0, 2, 1, 0, 5, 0, 0, 7, 8, 2, 10, 6, 13, 3, 15, 6, 12, 12, 10, 5, 1, 12, 8, 23, 7, 21, 14, 19, 29, 17, 16, 30, 6, 6, 33, 4, 1, 27, 35, 6, 4, 42, 39, 12, 35, 42, 43, 16, 3, 11, 14, 50, 33, 27, 47, 2, 30, 13, 50, 34, 43, 3, 63, 42, 2, 25, 13, 3, 8, 25, 20, 11, 42, 6, 27, 42, 38, 7, 20

Offset: 1

Author

G. C. Greubel, Nov 26 2018

Keywords

• sign

Comments

This expansion can also be considered as the expansion of -1/(golden ratio).

Examples

			(1-sqrt(5))/2 = -1 + 2/3! + 1/4! + 5/6! + 7/9! + 8/10! + 2/11! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for factorial base representation

Crossrefs

Cf. A001622, A068451, A094214 (decimal expansion, negated).

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]];

Mathematica

With[{b = -1/GoldenRatio}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

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)), ", "))

Sage

def a(n):
    if (n==1): return floor(-1/golden_ratio)
    else: return expand(floor(factorial(n)*(-1/golden_ratio)) - n*floor(factorial(n-1)*(-1/golden_ratio)))
[a(n) for n in (1..80)]