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 A270521 Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r(k) = 1/k!. 0
 2, 5, 10, 31, 359, 59268, 20093595288, 65918810162398275290, 1376774212248790880568204797925263397693, 867295417612705949534513217399066894914443846832286344139318038624107644813557 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. See A269993 for a guide to related sequences. LINKS Eric Weisstein's World of Mathematics, Egyptian Fraction EXAMPLE tau - 1 = 1/(1*2) + 1/(2*5) + 1/(6*10) + 1/(24*31) + ... MATHEMATICA r[k_] := 1/k!; f[x_, 0] = x; z = 10; n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]] f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k] x = -1 + GoldenRatio; Table[n[x, k], {k, 1, z}] PROG (PARI) r(k) = 1/k!; f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); ); a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016 CROSSREFS Cf. A269993, A000142, A094214. Sequence in context: A155580 A226963 A018386 * A089073 A343167 A138190 Adjacent sequences:  A270518 A270519 A270520 * A270522 A270523 A270524 KEYWORD nonn,frac,easy AUTHOR Clark Kimberling, Mar 30 2016 STATUS approved

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Last modified June 12 10:58 EDT 2021. Contains 344947 sequences. (Running on oeis4.)