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A228044 Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787. 6
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 (list; constant; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..86.

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]

CROSSREFS

Cf. A185787, A000027, A228040, A226985, A228045.

Sequence in context: A060804 A086364 A260662 * A171529 A260324 A157649

Adjacent sequences:  A228041 A228042 A228043 * A228045 A228046 A228047

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Aug 06 2013

STATUS

approved

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Last modified July 26 00:58 EDT 2017. Contains 289798 sequences.