A294828 - OEIS

%I #12 Nov 18 2017 03:43:20

%S 1,19,263,815,95597,678149,7531399,18016577,259695727,4173941423,

%T 222039686299,2153029760377,19428099753313,331021112488901,

%U 24211723390477517,12126560607807901,1008024074147249303,168229609596886043,1740462375346732831,1219642439745618215

%N Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 3) = A147874(k+2), for k >= 0.

%C The corresponding denominators are given in A294829.

%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] = [5,3].

%C The limit of the series is V(5,3) = lim_{n -> oo} V(5,3;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.48170177449... given in A294830.

%C In the Koecher reference v_5(3) = (2/5)*V(5,3) = 0.19268070979833151082... given there by ((1/4)*log(5) - (1/(2*sqrt(5)))*log((1+sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5)).

%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

%H Robert Israel, <a href="/A294828/b294828.txt">Table of n, a(n) for n = 0..891</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>

%F a(n) = numerator(V(5,3;n)) with V(5,3;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 3)) = Sum_{k=0..n} 1/A147874(k+2) = (1/2)*Sum_{k=0..n} (1/(k + 3/5) - 1/(k+1)) = (-Psi(3/5) + Psi(n+8/5) - (gamma + Psi(n+2)))/2 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

%e The rationals V(5,3;n), n >= 0, begin: 1/3, 19/48, 263/624, 815/1872, 95597/215280, 678149/1506960, 7531399/16576560, 18016577/39369330, 259695727/564293730, 4173941423/9028699680, 222039686299/478521083040, 2153029760377/4625703802720, 19428099753313/41631334224480, ...

%e V(5,3;10^6) = 0.4817015746 to be compared with 0.4817017745 from A294830 with 10 digits.

%p map(numer,ListTools:-PartialSums([seq(1/(k+1)/(5*k+3),k=0..50)])); # _Robert Israel_, Nov 17 2017

%o (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 3)))); \\ _Michel Marcus_, Nov 17 2017

%Y Cf. A001620, A001622, A147874, A294512, A294826/A294827 (V(5,2;n)), A294829, A294830,

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 16 2017