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A294512 Denominators of partial sums of the reciprocals of octagonal numbers. 11
1, 8, 168, 420, 5460, 14560, 276640, 3043040, 136936800, 136936800, 4245040800, 72165693600, 2670130663200, 2670130663200, 114815618517600, 1320379612952400, 9242657290666800, 3080885763555600, 280080523959600, 8122335194828400, 165154148961510800, 14533565108612950400, 973748862277067676800 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..22.

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

Cf. A000567, A244645, A250400, A250401.

Sequence in context: A305096 A032395 A316756 * A090228 A220808 A221022

Adjacent sequences:  A294509 A294510 A294511 * A294513 A294514 A294515

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 01 2017

STATUS

approved

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Last modified October 17 07:39 EDT 2021. Contains 348048 sequences. (Running on oeis4.)