A294513 Denominators of the partial sums of the reciprocals of twice the pentagonal numbers.

2, 5, 120, 1320, 9240, 52360, 52360, 602140, 70450380, 2043061020, 16344488160, 3268897632, 62109055008, 2546471255328, 1157486934240, 54401885909280, 272009429546400, 4805499921986400, 4805499921986400, 283524495397197600, 418536159872053600

Offset

0,1

Comments

The corresponding numerators are given by A250328(n+1), n >= 0.

Twice the positive pentagonal numbers are A049450(k+1) = (k+1)*(3*k+2), k >= 0.

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] = [3,2].

The limit of the series is V(3,2) = lim_{n -> oo} V(3,2;n) = (3/2)*log(3) - Pi/(2*sqrt(3)) = 0.74101875088505561179... given in A294514.

References

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193 (with v_m(r) = ((m-r)/m)*V(m,r)).

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = denominator(V(3,2;n)) with V(3,2;n) = Sum_{k=0..n} 1/((k + 1)*(3*k + 2)) = Sum_{k=0..n} 1/A049450(k+1) = Sum_{k=0..n} (3/(3*k + 2) - 1/(k+1)).

a(n) = 2*A250327(n+1)/(n+1) [conjecture].

Example

The rationals V(3,2;n), n >= 0, begin: 1/2, 3/5, 77/120, 877/1320, 6271/9240, 36049/52360, 36423/52360, 422137/602140, 49691099/70450380, 1448086909/2043061020, ...
V(3,2;10^4) = 0.7409854223(Maple, 10 digits) to be compared with 0.7410187513 from V(3,2) given in A294514.
Conjecture tests: a(0) = 2 =  A250327(1)/1, 2* a(1) = 5 = 2*A250327(2)/2 = A250327(2), a(2) = 120 = 2*A250327(2)/3 = 2*180/3, ...

Mathematica

Denominator@ Accumulate@ Array[1/(2 PolygonalNumber[5, #]) &, 21] (* Michael De Vlieger, Nov 02 2017 *)

Crossrefs

Cf. A049450, A250327(n+1), A250328(n+1), A294512.

Sequence in context: A221476 A015172 A132526 * A276109 A270481 A225944

Adjacent sequences: A294510 A294511 A294512 * A294514 A294515 A294516

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Nov 02 2017

Status

approved