A270521 - OEIS

%I #9 Nov 16 2024 23:52:37

%S 2,5,10,31,359,59268,20093595288,65918810162398275290,

%T 1376774212248790880568204797925263397693,

%U 867295417612705949534513217399066894914443846832286344139318038624107644813557

%N Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r(k) = 1/k!.

%C Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

%C See A269993 for a guide to related sequences.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e tau - 1 = 1/(1*2) + 1/(2*5) + 1/(6*10) + 1/(24*31) + ...

%t r[k_] := 1/k!; f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = -1 + GoldenRatio; Table[n[x, k], {k, 1, z}]

%o (PARI) r(k) = 1/k!;

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 31 2016

%Y Cf. A269993, A000142, A094214.

%K nonn,frac,easy,changed

%O 1,1

%A _Clark Kimberling_, Mar 30 2016