A228049 Decimal expansion of sum of reciprocals, column 3 of the natural number array, A185787.

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

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

1/4 + 1/8 + 1/13 + ... = (1/34)(17 + 8r*tan(r)), where r = (pi/2)sqrt(17)
1/4 + 1/8 + 1/13 + ... = 0.79841055101687800386526651756132658166...

Mathematica

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

Prog

(PARI) sumnumrat(2/(n^2+5*n+2), 1) \\ Charles R Greathouse IV, Feb 08 2023

Crossrefs

Cf. A185787, A000027, A228044, A226985.

Sequence in context: A348668 A259069 A209328 * A154943 A126041 A319881

Adjacent sequences: A228046 A228047 A228048 * A228050 A228051 A228052

Keyword

nonn,cons,easy

Author

Clark Kimberling, Aug 06 2013

Status

approved