A294514 -id:A294514 - cp's OEIS frontend

A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

1, 4, 8, 2, 0, 3, 7, 5, 0, 1, 7, 7, 0, 1, 1, 1, 2, 2, 3, 5, 9, 1, 6, 5, 7, 4, 5, 3, 1, 2, 5, 4, 2, 1, 3, 8, 1, 6, 5, 8, 4, 0, 5, 4, 2, 5, 3, 7, 5, 5, 0, 7, 7, 7, 9, 6, 3, 4, 1, 9, 8, 0, 6, 5, 5, 2, 4, 3, 5, 9, 6, 9, 8, 5, 2, 9, 4, 7, 3, 0, 1, 6, 9, 3, 6, 7, 2, 2, 2, 7, 6, 2, 2, 9, 1, 3, 6, 0, 9, 7, 5, 0, 7, 6, 8

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.482037501770111223591657453125421381658405425375507779634198065524359698529473...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Hongwei Chen and G. C. Greubel, Sum of the Reciprocals of Polygonal Numbers (Solved), SIAM Problems and solutions.
Hongwei Chen and G. C. Greubel, Siam, Problems and Solutions, problem 07-003 and the solution

Cf. A000326, A016650, A093602, A294514.

Decimal expansion of the sum of the reciprocals of the m-gonal numbers: A000038 (m=3), A013661 (m=4), this sequence (m=5), A016627 (m=6), A244639 (m=7), A244645 (m=8), A244646 (m=9), A244647 (m=10), A244648 (m=11), A244649 (m=12), A275792 (m=14).

SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // G. C. Greubel, Mar 24 2024

RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]
RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* G. C. Greubel, Mar 24 2024 *)

numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # G. C. Greubel, Mar 24 2024

Sum_{n>=1} 2/(3*n^2 - n).
Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - Michel Marcus, Jul 03 2014
Equals 2*A294514. - Hugo Pfoertner, Apr 24 2025

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

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 02 2017

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.

			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, ...

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

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

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

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].

A384683 Decimal expansion of Sum_{i >= 1} 1/(3i-1) - 1/(3i).

Original entry on oeis.org

2, 4, 7, 0, 0, 6, 2, 5, 0, 2, 9, 5, 0, 1, 8, 5, 3, 7, 2, 6, 5, 2, 7, 6, 2, 4, 2, 1, 8, 7, 5, 7, 0, 2, 3, 0, 2, 7, 6, 4, 0, 0, 9, 0, 4, 2, 2, 9, 2, 5, 1, 2, 9, 6, 6, 0, 5, 6, 9, 9, 6, 7, 7, 5, 8, 7, 3, 9, 3, 2, 8, 3, 0, 8, 8, 2, 4, 5, 5, 0, 2, 8, 2, 2, 7, 8, 7, 0, 4, 6, 0, 3, 8, 1, 8, 9, 3, 4, 9, 5, 8, 4, 6, 1, 4, 6, 1, 2, 1, 1, 9, 4, 6, 7, 8, 4

Offset: 0

Jason Bard, Jun 06 2025

Generalization of infinite sum generating A002162 (natural logarithm of 2). That sum is Sum_{i >= 1} 1/(k*i-1) - 1/(k*i), where k = 2. Here, we set k = 3.

			0.24700625029501853726527624218757023027640090422925...

Jason Bard, Table of n, a(n) for n = 0..9999
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 291.
Index entries for transcendental numbers.

Cf. A002162, A007494, A152743, A156057, A294514, A381671.

RealDigits[1/2 * Log[3] - (Sqrt[3]/18) * Pi, 10, 1000][[1]]

log(3)/2 - Pi/(6*sqrt(3)) \\ Amiram Eldar, Jun 07 2025

Equals (1/2) * log(3) - sqrt(3) * Pi / 18.
Equals Sum_{i>=1} 1/A152743(i).
Equals A294514/3. - Hugo Pfoertner, Jun 07 2025

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A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

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

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A384683 Decimal expansion of Sum_{i >= 1} 1/(3i-1) - 1/(3i).

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cp's OEIS Frontend

A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

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

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A384683 Decimal expansion of Sum_{i >= 1} 1/(3*i-1) - 1/(3*i).

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A384683 Decimal expansion of Sum_{i >= 1} 1/(3i-1) - 1/(3i).