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A270744 (r,1)-greedy sequence, where r(k) = 1/tau^k, where tau = golden ratio.

%I A270744 #10 Mar 07 2025 01:28:11
%S A270744 1,2,2,3,4,32,1065,2038968,5977146319204,36314862033946243071181679,
%T A270744 1028280647188781709727717632740627249617427013751977,
%U A270744 958046899855070460620234639622630375078362220775180051610386376308132568342498992099474472596860400289
%N A270744 (r,1)-greedy sequence, where r(k) = 1/tau^k, where tau = golden ratio.
%C A270744 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.  (The same algorithm is used to generate sequences listed at A269993.)
%C A270744 Guide to related sequences:
%C A270744   x     r(k)
%C A270744   1   1/tau^k             A270744
%C A270744   1   k/tau^k             A270745
%C A270744   1   2/e^k               A270746
%C A270744   1   4/Pi^k              A270747
%C A270744   1   2/log(k+1)          A270748
%C A270744   1   k/log(k+1)          A270749
%C A270744   1   1/(k*log(k+1))      A270750
%C A270744   1   1/(k*tau)           A270751
%C A270744   1   1/(k*e)             A270752
%C A270744   1   1/(k*sqrt(2))       A270916
%F A270744 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 A270744 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%e A270744 a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
%e A270744 a(2) = ceiling(r(2)/(1 - r(1)/1)) = 2;
%e A270744 a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2)) = 2.
%e A270744 The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.618..., 0.809..., 0.927..., 0.975..., 0.998..., 0.999967... .
%t A270744 $MaxExtraPrecision = Infinity; z = 13;
%t A270744 r[k_] := N[1/GoldenRatio^k, 1000]; f[x_, 0] = x;
%t A270744 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270744 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270744  x = 1; Table[n[x, k], {k, 1, z}]
%t A270744 N[Sum[r[k]/n[x, k], {k, 1, 13}], 200]
%Y A270744 Cf. A001620, A270745, A094214, A269993.
%K A270744 nonn,easy
%O A270744 1,2
%A A270744 _Clark Kimberling_, Apr 07 2016