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%I #12 Sep 08 2022 08:46:20
%S 1,15,599,23035,2900123,30112021,1117973277,96393597197,6084978910411,
%T 67042215785861,4094947551504521,274661892011507657,
%U 20068897076286721961,1586702257063428405419,26992510145660626515763,54017546409271099350401,5242487768036648180534897,180077149085745155963315797
%N Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)*(6*k + 1) = A051866(k+1).
%C The corresponding denominators are given in A294835.
%C For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [6,1].
%C The limit of the series is V(6,1) = lim_{n -> oo} V(6,1;n) = (3/10)*log(3) + (2/5)*log(2) + (1/10)*Pi*sqrt(3). The value is 1.150982368094676386... given in A275792.
%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
%H G. C. Greubel, <a href="/A294834/b294834.txt">Table of n, a(n) for n = 0..600</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>
%F a(n) = numerator(V(6,1;n)) with V(6,1;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 1)) = Sum_{k=0..n} 1/A051866(k+1) = (1/5)*Sum_{k=0..n} (1/(k + 1/6) - 1/(k + 1)) = (-Psi(1/6) + Psi(n+7/6) - (gamma + Psi(n+2)))/5 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.
%e The rationals V(6,1;n), n >= 0, begin: 1, 15/14, 599/546, 23035/20748, 2900123/2593500, 30112021/26799500, 1117973277/991581500, 96393597197/85276009000, 6084978910411/5372388567000, 67042215785861/59096274237000, 4094947551504521/3604872728457000, ...
%e V(6,1;10^6) = 1.150982200 (Maple, 10 digits) to be compared with the ten digits 1.150982368 obtained from V(6,1) given in A275792.
%t Table[Numerator[Sum[1/((k + 1)*(6*k + 1)), {k, 0, n}]], {n, 0, 50}] (* _G. C. Greubel_, Aug 30 2018 *)
%o (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 1)))); \\ _Michel Marcus_, Nov 21 2017
%o (Magma) [Numerator((&+[1/((k + 1)*(6*k + 1)): k in [0..n]])): n in [0..50]]; // _G. C. Greubel_, Aug 30 2018
%Y Cf. A001620, A051866, A275792, A294512, A294835.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 20 2017
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