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A270751 (r,1)-greedy sequence, where r(k) = 1/(k*tau) and tau = golden ratio.

%I A270751 #14 Jun 21 2025 14:48:15
%S A270751 1,1,3,37,1204,21029921,425355555167420,
%T A270751 439183524292095499600664584581,
%U A270751 240317442633783387248198509182959563857071128274317237128901
%N A270751 (r,1)-greedy sequence, where r(k) = 1/(k*tau) and tau = golden ratio.
%C A270751 Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers.  Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x.  The sequence (a(n)) is the (r,x)-greedy sequence.  We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x.  See A270744 for a guide to related sequences.
%F A270751 a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).
%F A270751 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%F A270751 Conjecture: a(n) = A270584(n-1) for n>1. - _R. J. Mathar_, Jun 21 2025
%e A270751 a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
%e A270751 a(2) = ceiling(r(2)/(1 - r(1)/1)) = 1;
%e A270751 a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/1)) = 3.
%e A270751 The first 3 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.618..., 0.927..., 0.995...
%t A270751 $MaxExtraPrecision = Infinity; z = 16;
%t A270751 r[k_] := N[1/(k*GoldenRatio), 1000]; f[x_, 0] = x;
%t A270751 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270751 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270751 x = 1; Table[n[x, k], {k, 1, z}]
%t A270751 N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]
%Y A270751 Cf. A001620, A270744, A094214.
%K A270751 nonn,easy
%O A270751 1,3
%A A270751 _Clark Kimberling_, Apr 09 2016