A270744 - OEIS

A270744

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

%I #4 Apr 09 2016 16:27:34

%S 1,2,2,3,4,32,1065,2038968,5977146319204,36314862033946243071181679,

%T 1028280647188781709727717632740627249617427013751977,

%U 958046899855070460620234639622630375078362220775180051610386376308132568342498992099474472596860400289

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

%C 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 Guide to related sequences:

%C x r(k)

%C 1 1/tau^k A270744

%C 1 k/tau^k A270745

%C 1 2/e^k A270746

%C 1 4/Pi^k A270747

%C 1 2/log(k+1) A270748

%C 1 1/(k*log(k+1)) A270749

%C 1 (1/k)*log(k+1) A270750

%C 1 2/(k*tau^k) A270751

%C 1 1/(k*e) A270752

%C 1 1/(k*sqrt(2)) A270916

%F 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 r(1)/a(1) + ... + r(n)/a(n) + ... = 1

%e a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;

%e a(2) = ceiling(r(2)/(1 - r(1)/1) = 2;

%e a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2) = 2.

%e The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are

%e 0.618..., 0.809..., 0.927..., 0.975..., 0.998..., 0.999967... .

%t $MaxExtraPrecision = Infinity; z = 13;

%t r[k_] := N[1/GoldenRatio^k, 1000]; f[x_, 0] = x;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = 1; Table[n[x, k], {k, 1, z}]

%t N[Sum[r[k]/n[x, k], {k, 1, 13}], 200]

%Y Cf. A001620, A270745, A094214, A269993.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Apr 07 2016