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A228044
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Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787.
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6
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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...
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MATHEMATICA
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$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]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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