OFFSET
0,2
COMMENTS
The corresponding numerators are given in A250401.
The octagonal numbers are here A000567(k+1) = (k + 1)*(3*k + 1), k >= 0.
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).
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.)
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.
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.
FORMULA
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.
a(n) = A250400(n+1)/(n+1), n >= 0. [conjecture].
EXAMPLE
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, ...
V(3,1,10^4) = 1.2773757281147540626 (Maple 20 digits) to be compared with 1.2774090575596367312 (20 digits from A244645).
The series is V(3,1) = 1 + 1/(2*4) + 1/(3*6) + 1/(4*10) + ... .
MATHEMATICA
Denominator@ Accumulate@ Array[1/PolygonalNumber[8, #] &, 23] (* Michael De Vlieger, Nov 01 2017 *)
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 01 2017
STATUS
approved