A228044 Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787.

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

Offset: 1

Author

Clark Kimberling, Aug 06 2013

Keywords

• nonn
• cons
• easy

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(k) be the sum of reciprocals of the numbers in column k 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.

It appears that Mathematica gives closed-form exact expressions for s(n), c(n) for 1<=n<=20 and further. The same holds for diagonal sums dr(n,n+k), k>=0; and for diagonal sums and dc(n+k,n), k>=0. In any case, general terms for all four sequences can be represented using the digamma function. The representations imply that c(n) is rational if and only if n is a term of A000124, and that dr(n) is rational if and only if n has the form k^2 + 2 for k >= 1.

Examples

			1.12229460660350434354343218597925...
		

Crossrefs

Cf. A185787, A000027, A228040, A226985, A228045.

Programs

Mathematica

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

PARI

sumnumrat(2/(n*(n+1)+4),1) \\ Charles R Greathouse IV, Feb 08 2023

Formula

Equals 1/3 + 1/5 + 1/8 + ...

Equals (1/30)*(-15 + 8*r*tanh(r)), where r = (Pi/2)*sqrt(15).