A228045 Decimal expansion of sum of reciprocals, row 3 of the natural number array, A185787.

7, 2, 6, 8, 0, 0, 6, 1, 9, 4, 6, 4, 9, 3, 5, 7, 7, 8, 1, 7, 9, 1, 4, 3, 0, 0, 7, 1, 9, 4, 4, 3, 5, 3, 8, 3, 9, 0, 8, 7, 7, 4, 6, 2, 7, 6, 3, 6, 0, 7, 7, 7, 3, 2, 3, 8, 9, 9, 7, 2, 6, 1, 3, 4, 0, 1, 3, 4, 6, 7, 2, 7, 2, 0, 1, 4, 8, 5, 9, 5, 2, 6, 4, 2, 6, 4

Offset

0,1

Comments

Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047.

Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048.

Links

Table of n, a(n) for n=0..85.

Example

1/6 + 1/9 + 1/13 + ... = (1/276)*(-161 + 48r*tanh(r/2), where r=(pi/2)sqrt(23).
1/6 + 1/9 + 1/13 + ... = 0.726800619464935778179143007194435...

Mathematica

$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[3, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]

Crossrefs

Cf. A185787, A000027, A228044, A226985.

Sequence in context: A322933 A143306 A199075 * A011361 A199433 A095398

Adjacent sequences: A228042 A228043 A228044 * A228046 A228047 A228048

Keyword

nonn,cons,easy

Author

Clark Kimberling, Aug 06 2013

Status

approved