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A244639 Decimal expansion of the sum of the reciprocals of the heptagonal numbers (A000566). 9
1, 3, 2, 2, 7, 7, 9, 2, 5, 3, 1, 2, 2, 3, 8, 8, 8, 5, 6, 7, 4, 9, 4, 4, 2, 2, 6, 1, 3, 1, 0, 0, 8, 4, 0, 1, 6, 5, 2, 2, 8, 0, 1, 1, 7, 3, 7, 1, 3, 9, 2, 4, 3, 7, 2, 2, 8, 5, 4, 5, 7, 6, 2, 6, 8, 8, 5, 1, 6, 2, 2, 1, 0, 7, 6, 8, 5, 8, 4, 4, 7, 5, 3, 5, 6, 8, 0, 9, 0, 8, 6, 0, 4, 1, 2, 4, 4, 7, 1, 1, 9, 3, 2, 0, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
For the partial sums of one half of this series, that is Sum_{k>=0} 1/((k+1)*(5*k+2)), with value 0.6613896265611944283..., see A294826(n)/A294827(n), for n >= 0. - Wolfdieter Lang, Nov 16 2017
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
LINKS
L. Downey, B. W. Ong, and J. A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J. 39, no. 8, 2008, 391-394.
Society for Industrial and Applied Mathematics, Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss
FORMULA
Equals Sum_{n>=1} 2/(5n^2 - 3n).
((5/2)*log(5) - (2*phi-1)*(log(phi) - (Pi/5)*sqrt(7-4*phi)))/3, with the golden section phi := (1 + sqrt(5))/2. This is (5/10)*v_5(2) given from the Koecher reference on p. 192 as ((5/2)*log(5) - sqrt(5)*log((1+sqrt(5))/2) + (1/5)*Pi*sqrt(5*(5-2*sqrt(5))))/3. Compare this with the number given in the Mathematica program. - Wolfdieter Lang, Nov 16 2017
EXAMPLE
1.32277925312238885674944226131008401652280117371392437228545762688516221076....
MATHEMATICA
RealDigits[ Pi*Sqrt[25 - 10 Sqrt[5]]/15 + 2Log[5]/3 + (1 + Sqrt[5]) Log[ Sqrt[ 10 - 2 Sqrt[5]]/2]/3 + (1 - Sqrt[5]) Log[ Sqrt[ 10 + 2 Sqrt[5]]/2]/3, 10, 111][[1]] (* or *)
RealDigits[ Sum[2/(5 n^2 - 3 n), {n, 1, Infinity}], 10, 111][[1]]
PROG
(PARI) sumnumrat(2/n/(5*n-3), 1) \\ Charles R Greathouse IV, Feb 08 2023
CROSSREFS
Sequence in context: A092419 A293268 A020835 * A352673 A193919 A055674
KEYWORD
nonn,cons,easy
AUTHOR
Robert G. Wilson v, Jul 03 2014
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)