A101629 Numerator of partial sums of a certain series.

1, 47, 6931, 238657, 4563655, 526760263, 45934377581, 2852342564497, 105651280880749, 4335127472172929, 186521117762900387, 61393482232562091673, 3255023127143379846869, 3255958701070954680689

Offset

1,2

Comments

The denominators are given in A101630.

Third member (m=4) of a family defined in A101028.

The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 15*sum(Zeta(2*k+1)/4^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 15*(3*log(2)-2) = 1.1916231251...; see the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/4 together with p. 258.

Links

Table of n, a(n) for n=1..14.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

W. Lang: Rationals s(n) and more.

Formula

a(n)=numerator(s(n)) with s(n)=60*sum(1/((4*k-1)*(4*k)*(4*k+1)), k=1..n) = 15*sum(1/((4*k-1)*k*(4*k+1)), k=1..n).

Example

s(3)= 60*(1/(3*4*5)+ 1/(7*8*9) + 1/(11*12*13)) = 6931/6006, hence
a(3)=6931 and A101630(3)=6006.

Crossrefs

Cf. A101028, A101627, A101631, members 2, 3, 5, resp.

Sequence in context: A217230 A217231 A141797 * A362714 A267063 A287225

Adjacent sequences: A101626 A101627 A101628 * A101630 A101631 A101632

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Dec 23 2004

Status

approved