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 A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+3) = A147874(k+2) for k >= 0. 4
 4, 8, 1, 7, 0, 1, 7, 7, 4, 4, 9, 5, 8, 2, 8, 7, 7, 7, 0, 7, 7, 0, 7, 5, 9, 2, 9, 3, 6, 1, 9, 1, 4, 7, 5, 5, 2, 3, 4, 1, 8, 7, 4, 5, 9, 3, 7, 4, 8, 4, 1, 8, 0, 4, 7, 3, 0, 4, 5, 9, 0, 1, 4, 1, 8, 8, 1, 5, 0, 5, 5, 7, 2, 3, 1, 7, 1, 8, 8, 9, 7, 5, 6, 8, 1, 9, 7, 7, 0, 2, 2, 1, 4, 0, 1, 6, 0, 3, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS In the Koecher reference v_5(3) = (2/5)*(present value V(5,3)) = 0.192680709798338..., given there as (1/4)*log(5) - (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5)). REFERENCES Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193. LINKS G. C. Greubel, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Digamma Function FORMULA Sum_{k>=0} 1/((5*n + 3)*(n + 1)) =: V(5,3) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 = (1/2)*(-Psi(3/5) + Psi(1)) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620. The partial sums of this series are given in A294828/A294829. EXAMPLE 0.481701774495828777077075929361914755234187459374841804730459014188150... MATHEMATICA RealDigits[((5/2)*Log[5] - (2*GoldenRatio - 1)*(Log[GoldenRatio] + (Pi/5)*Sqrt[7 - 4*GoldenRatio]))/4, 10, 100][[1]] (* G. C. Greubel, Aug 30 2018 *) PROG (PARI) default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 \\ G. C. Greubel, Aug 30 2018 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); phi:= (1 + Sqrt(5))/2; ((5/2)*Log(5) - (2*phi-1)*(Log(phi) + (Pi(R)/5)*Sqrt(7 - 4*phi)))/4; // G. C. Greubel, Aug 30 2018 CROSSREFS Cf. A001620, A001622, A147874, A294828/A294829. Sequence in context: A372805 A343058 A196618 * A248415 A328250 A340423 Adjacent sequences: A294827 A294828 A294829 * A294831 A294832 A294833 KEYWORD nonn,cons AUTHOR Wolfdieter Lang, Nov 16 2017 STATUS approved

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Last modified August 11 14:19 EDT 2024. Contains 375071 sequences. (Running on oeis4.)