|
%I #10 Feb 08 2023 16:38:32
%S 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,
%T 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,
%U 7,8,3,0,9,5,1,4,3,1,1,1,3,3,5,4,1,9
%N Decimal expansion of sum of reciprocals, column 3 of the natural number array, A185787.
%C 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.
%C 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.
%e 1/4 + 1/8 + 1/13 + ... = (1/34)(17 + 8r*tan(r)), where r = (pi/2)sqrt(17)
%e 1/4 + 1/8 + 1/13 + ... = 0.79841055101687800386526651756132658166...
%t $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]
%o (PARI) sumnumrat(2/(n^2+5*n+2),1) \\ _Charles R Greathouse IV_, Feb 08 2023
%Y Cf. A185787, A000027, A228044, A226985.
%K nonn,cons,easy
%O 0,1
%A _Clark Kimberling_, Aug 06 2013
|
