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%I #13 Nov 12 2017 08:11:55
%S 1,8,168,420,5460,14560,276640,3043040,136936800,136936800,4245040800,
%T 72165693600,2670130663200,2670130663200,114815618517600,
%U 1320379612952400,9242657290666800,3080885763555600,280080523959600,8122335194828400,165154148961510800,14533565108612950400,973748862277067676800
%N Denominators of partial sums of the reciprocals of octagonal numbers.
%C The corresponding numerators are given in A250401.
%C The octagonal numbers are here A000567(k+1) = (k + 1)*(3*k + 1), k >= 0.
%C In general the partial sums 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 are of interest for series Sum_{k>=1} a(k)/k with a periodic sequence a(r + m*k) = a(r), r = 1..m, k >= 1, and Sum_{r=1..m} a(r) = 0. Such sequences were considered by Euler in his Introductio in Analysin Infinitorum (1748). See the Koecher reference. Namely, Sum_{k>=1} a(k)/k = Sum_{r=1..m-1} a(r)*v_m(r) with v_m(r) = ((m-r)/m)*lim_{n -> oo} V(m,r,n).
%C The general formula is m*v_m(r) = log(m) + (Pi/2)*cot(Pi*r/m) - Sum_{s=1..m-1} cos(2*Pi*r*s/m)*log(2*sin((Pi*s)/m)), r = 1..m-1. (Koecher, Satz, p. 191.)
%C Here the instance m = 3, r = 1 is considered with V(3,1;n) = Sum_{k=0..n} 1/((k + 1)*(3*k + 1)) and lim_{n -> oo} V(3,1;n) = (Pi/sqrt(3) + 3*log(3))/4 with its decimal expansion 1.277409057... given in A244645.
%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.
%F a(n) = denominator(V(3,1;n)) with V(3,1;n) = Sum_{k=0..n} 1/((k + 1)*(3*k + 1)) = (1/2)*Sum_{k=0..n} (3/(3*k + 1) - 1/(k+1)), n >= 0.
%F a(n) = A250400(n+1)/(n+1), n >= 0. [conjecture].
%e The rationals V(3,1;n) begin: 1, 9/8, 197/168, 503/420, 6623/5460, 17813/14560, 340527/276640, 3763087/3043040, 169947523/136936800, 170436583/136936800, ...
%e V(3,1,10^4) = 1.2773757281147540626 (Maple 20 digits) to be compared with 1.2774090575596367312 (20 digits from A244645).
%e The series is V(3,1) = 1 + 1/(2*4) + 1/(3*6) + 1/(4*10) + ... .
%t Denominator@ Accumulate@ Array[1/PolygonalNumber[8, #] &, 23] (* _Michael De Vlieger_, Nov 01 2017 *)
%Y Cf. A000567, A244645, A250400, A250401.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 01 2017
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