This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2017 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 16:41 EST 2018. Contains 318023 sequences. (Running on oeis4.)