A294826 - OEIS

%I #24 Aug 31 2023 17:29:49

%S 1,4,151,1315,36698,667109,10749479,399851303,401511863,18933826729,

%T 246810236317,4700047812703,145981746528913,9796912235587651,

%U 9810925971351679,9823210739716249,403196782523223569,11704197956499986461,269433333504358946963,5231145593209503407215,747842028258712790473

%N Numerators of the partial sums of the reciprocals of twice the heptagonal numbers (k + 1)*(5*k + 2) = A135706(k+1) = 2*A000566(k+1), for k >= 0.

%C The corresponding denominators are given in A294827.

%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] = [5,2].

%C The limit of the series is V(5,2) = lim_{n -> oo} V(5,2;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) - (Pi/5)*sqrt(7-4*phi)))/6, with the golden section phi:= (1 + sqrt(5))/2. The value is 0.661389626561... given by (1/2)*A244639.

%C In the Koecher reference v_5(2) =  (3/5)*V(5,2) = 0.39683377593671665701 ...is given as (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 G. C. Greubel, <a href="/A294826/b294826.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(5,2;n)) with V(5,2;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 2)) = Sum_{k=0..n} 1/A135706(k+1) = (1/3)*Sum_{k=0..n} (1/(k + 2/5) - 1/(k+1)) = (-Psi(2/5) + Psi(n+7/5) - (gamma + Psi(n+2)))/3 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

%e The rationals V(5,2;n), n >= 0, begin: 1/2, 4/7, 151/252, 1315/2142, 36698/58905, 667109/1060290, 10749479/16964640, 399851303/627691680, 401511863/627691680, 18933826729/29501508960, 246810236317/383519616480, ...

%e V(5,2;10^6) = 0.6613894266 (Maple, 10 digits) to be compared with 0.6613896266 giving the 10 digit value of V(5,2) from (1/2)*A244649.

%t Table[Numerator[Sum[1/((k+1)*(5*k+2)), {k,0,n}]], {n,0,25}] (* _G. C. Greubel_, Aug 29 2018 *)

%t Accumulate[1/(2*PolygonalNumber[7,Range[30]])]//Numerator (* _Harvey P. Dale_, Aug 31 2023 *)

%o (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 2)))); \\ _Michel Marcus_, Nov 17 2017

%o (Magma) [Numerator((&+[1/((k+1)*(5*k+2)): k in [0..n]])): n in [0..25]]; // _G. C. Greubel_, Aug 29 2018

%Y Cf. A001620, A000566, A135706, A294512, A294520/A294521 (V(5,1;n)), A244639.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 16 2017