|
%I #11 Feb 08 2023 15:39:23
%S 4,2,3,5,3,9,0,9,9,6,0,8,7,0,0,1,9,6,8,3,7,6,0,7,6,8,9,9,7,4,4,2,8,9,
%T 3,7,5,4,4,3,2,2,8,8,1,8,9,4,1,7,1,1,1,0,2,1,7,5,6,0,8,4,2,8,1,3,0,9,
%U 3,4,7,8,2,4,5,8,2,6,7,1,1,7,8,2,5,9
%N Decimal expansion of sum of reciprocals, row 5 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 1/15 + 1/20 + 1/26 + ... = (1/17160)(-9997 + 1760r*tanh(r)), where r = (pi/2)*sqrt(39)
%e 1/15 + 1/20 + 1/26 + ... = 0.42353909960870019683760768997442893...
%t $MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[5, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]
%o (PARI) sumnumrat(1/(n*(n+7)/2+11),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
|
