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A270745 (r,1)-greedy sequence, where r(k) = k/tau^k and tau = golden ratio.

%I A270745 #13 Mar 07 2025 01:28:14
%S A270745 1,3,6,63,18292,822680526,810795400558456607,
%T A270745 3059608561526409532580548135921262544,
%U A270745 41330361304323764432512049819825720283867354602583921458622085452146872094
%N A270745 (r,1)-greedy sequence, where r(k) = k/tau^k and tau = golden ratio.
%C A270745 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 A270745 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 A270745 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%e A270745 a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
%e A270745 a(2) = ceiling(r(2)/(1 - r(1)/1)) = 3;
%e A270745 a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/3)) = 6.
%e A270745 The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.618..., 0.872..., 0.990..., 0.975..., 0.999..., 0.99999999959...
%t A270745 $MaxExtraPrecision = Infinity; z = 13;
%t A270745 r[k_] := N[k/GoldenRatio^k, 1000]; f[x_, 0] = x;
%t A270745 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270745 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270745 x = 1; Table[n[x, k], {k, 1, z}]
%t A270745 N[Sum[r[k]/n[x, k], {k, 1, 13}], 200]
%Y A270745 Cf. A001620, A270744, A094214.
%K A270745 nonn,easy
%O A270745 1,2
%A A270745 _Clark Kimberling_, Apr 09 2016