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A228045
Decimal expansion of sum of reciprocals, row 3 of the natural number array, A185787.
1
7, 2, 6, 8, 0, 0, 6, 1, 9, 4, 6, 4, 9, 3, 5, 7, 7, 8, 1, 7, 9, 1, 4, 3, 0, 0, 7, 1, 9, 4, 4, 3, 5, 3, 8, 3, 9, 0, 8, 7, 7, 4, 6, 2, 7, 6, 3, 6, 0, 7, 7, 7, 3, 2, 3, 8, 9, 9, 7, 2, 6, 1, 3, 4, 0, 1, 3, 4, 6, 7, 2, 7, 2, 0, 1, 4, 8, 5, 9, 5, 2, 6, 4, 2, 6, 4
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.
EXAMPLE
1/6 + 1/9 + 1/13 + ... = (1/276)*(-161 + 48r*tanh(r/2), where r=(pi/2)sqrt(23).
1/6 + 1/9 + 1/13 + ... = 0.726800619464935778179143007194435...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[3, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]
CROSSREFS
KEYWORD
nonn,cons,easy
AUTHOR
Clark Kimberling, Aug 06 2013
STATUS
approved