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Factorial expansion of (1-sqrt(5))/2 = Sum_{n>=1} a(n)/n!.
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%I #18 May 24 2022 00:11:22

%S -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,

%T 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,

%U 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

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

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

%H G. C. Greubel, <a href="/A322119/b322119.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation</a>

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

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

%o (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)), ", "))

%o (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]];

%o (Sage)

%o def a(n):

%o if (n==1): return floor(-1/golden_ratio)

%o else: return expand(floor(factorial(n)*(-1/golden_ratio)) - n*floor(factorial(n-1)*(-1/golden_ratio)))

%o [a(n) for n in (1..80)]

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

%K sign

%O 1,3

%A _G. C. Greubel_, Nov 26 2018