A070986 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k^3.

1, 2, 5, 9, 14, 10, 15, 16, 14, 20, 22, 19, 33, 25, 27, 31, 41, 33, 52, 47, 36, 54, 38, 56, 50, 56, 65, 68, 81, 76, 77, 84, 75, 80, 88, 76, 90, 89, 98, 89, 113, 100, 110, 101, 104, 103, 121, 122, 131, 121, 131, 127, 124, 134, 127, 132, 143, 131, 143, 144, 161, 141

Offset: 1

Author

Benoit Cloitre, May 18 2002

Keywords

• easy
• nonn

Comments

Sum_{k>=1} 1/k^3 = zeta(3) = 1.2020... (A002117).

Examples

			The simple continued fraction for Sum_{k=1..10} 1/k^3 is [1, 5, 16, 135, 1, 5, 2, 3, 2, 1, 1, 1, 3, 3, 16, 1, 2, 1, 1, 2] which contains 20 terms, hence a(10)=20.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002117, A055573.

Programs

Mathematica

lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k^3, {k, 1, 100}]] (* Amiram Eldar, Apr 29 2022 *)

PARI

for(n=1,100,print1(length(contfrac(sum(i=1,n,1/i^3))),","))

Formula

Limit_{n ->infinity} a(n)/n = C = 2.5....