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A270351 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1, 1/2, 1/4, 1/8, ...)

%I A270351 #15 Feb 16 2025 08:33:31
%S A270351 2,5,14,707,1470654,1143462781221,1805535113251940020114035,
%T A270351 2497859054491311040375647235065337168455108737151,
%U A270351 3189945744303964831068292153370103839290925070278698110007359838830245675325591867634500100743606
%N A270351 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1, 1/2, 1/4, 1/8, ...)
%C A270351 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 A270351 See A269993 for a guide to related sequences.
%H A270351 Clark Kimberling, <a href="/A270351/b270351.txt">Table of n, a(n) for n = 1..11</a>
%H A270351 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A270351 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A270351 tau - 1 = 1/2 + 1/(2*5) + 1/(4*14) + ...
%t A270351 r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
%t A270351 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270351 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270351 x = GoldenRatio; Table[n[x, k], {k, 1, z}]
%o A270351 (PARI) r(k) = 2/2^k;
%o A270351 f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
%o A270351 a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 18 2016
%Y A270351 Cf. A269993.
%K A270351 nonn,frac,easy
%O A270351 1,1
%A A270351 _Clark Kimberling_, Mar 17 2016