A270747 (r,1)-greedy sequence, where r(k) = 4/Pi^k.

2, 2, 1, 2, 2, 1, 3, 7, 71, 3131, 5821925, 14364035515947, 451397201144015321568515204, 88020328073777548345010277436911545872870466008026310, 4344173888544359227731947461270153179826227998155726069662805370800638822815760136590246135744249701337368

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..15.

Formula

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).

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

Example

a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 2;
a(2) = ceiling(r(2)/(1 - r(1)/1) = 2;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2) = 1.
The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ...  are
0.636..., 0.839..., 0.968..., 0.988..., 0.995..., 0.9994...

Mathematica

$MaxExtraPrecision = Infinity; z = 16;
r[k_] := N[4/Pi^k, 1000]; f[x_, 0] = x;
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; Table[n[x, k], {k, 1, z}]
N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]

Crossrefs

Cf. A001620, A270744, A049541.

Sequence in context: A023671 A302243 A117535 * A214517 A230594 A072463

Adjacent sequences: A270744 A270745 A270746 * A270748 A270749 A270750

Keyword

nonn,easy

Author

Clark Kimberling, Apr 09 2016

Status

approved