A096949 Numerators of partial sums of series for 3*arctanh(1/3) = (3/2)*log(2).

1, 28, 421, 26528, 2148803, 7878956, 2765513941, 74668877408, 3808112752813, 651187280816108, 2511722368895123, 173308843453994432, 7798897955430811787, 1895132203169713822916, 54958833891921780540589

Offset

0,2

Comments

Denominators are given in A096950.

From the series of log((1+x)/(1-x)) for x = 1/3, i.e., for log(2) = 2*Sum_{k>=0} (1/3)^(2*k+1)/(2*k+1).

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = numerator(A(n)) with the rational number A(n) := Sum_{k=0..n} (1/3)^(2*k)/(2*k+1) in lowest terms.

(3/2)*log(2) = a(n)/A096950(n) + 3*Integral_{x >= 3} 1/(x^(2*n+4) - x^(2*n+2)) dx. - Peter Bala, Feb 05 2024

Example

n=3: 26528/A096950(3) = 26528/25515 = 1.0397021... approximates 3*arctanh(1/3) = 1.039720771...
n = 3: Sum_{k = 0..3} (1/3)^(2*k + 1)/(2*k + 1) = 1/3 + 1/81 + 1/1215 + 1/15309 = 26528/76545. Hence a(3) = 26528. - Peter Bala, Feb 06 2024

Crossrefs

Cf. A096950.

Sequence in context: A271793 A024213 A004415 * A093974 A121803 A022656

Adjacent sequences: A096946 A096947 A096948 * A096950 A096951 A096952

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jul 16 2004

Status

approved