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

%I A322119 #18 May 24 2022 00:11:22
%S A322119 -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 A322119 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 A322119 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 A322119 Factorial expansion of (1-sqrt(5))/2 = Sum_{n>=1} a(n)/n!.
%C A322119 This expansion can also be considered as the expansion of -1/(golden ratio).
%H A322119 G. C. Greubel, <a href="/A322119/b322119.txt">Table of n, a(n) for n = 1..10000</a>
%H A322119 <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation</a>
%e A322119 (1-sqrt(5))/2 = -1 + 2/3! + 1/4! + 5/6! + 7/9! + 8/10! + 2/11! + ...
%t A322119 With[{b = -1/GoldenRatio}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]
%o A322119 (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 A322119 (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 A322119 (Sage)
%o A322119 def a(n):
%o A322119     if (n==1): return floor(-1/golden_ratio)
%o A322119     else: return expand(floor(factorial(n)*(-1/golden_ratio)) - n*floor(factorial(n-1)*(-1/golden_ratio)))
%o A322119 [a(n) for n in (1..80)]
%Y A322119 Cf. A001622, A068451, A094214 (decimal expansion, negated).
%K A322119 sign
%O A322119 1,3
%A A322119 _G. C. Greubel_, Nov 26 2018