OFFSET
1,3
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.
EXAMPLE
1/3 + 1/5 + 1/8 + ... = (1/30)*(-15 + 8r*tanh(r/2), where r=(pi/2)sqrt(15).
1/3 + 1/5 + 1/8 + ... = 1.12229460660350434354343218597925...
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]
PROG
(PARI) sumnumrat(2/(n*(n+1)+4), 1) \\ Charles R Greathouse IV, Feb 08 2023
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Aug 06 2013
STATUS
approved