A244648 -id:A244648 - cp's OEIS frontend

A244645 Decimal expansion of the sum of the reciprocals of the octagonal numbers (A000567).

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.2774090575596367311949534921024332115566344803902472326934919840751515151955452...

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Wikipedia, Polygonal number

Cf. A000038, A013661, A244639, A244644, A244646, A244647, A244648, A244649, A000567.

RealDigits[ Sum[1/(3n^2 - 2n), {n, 1 , Infinity}], 10, 111][[1]]

sumpos(n=1, 1/(3*n^2 - 2*n)) \\ Michel Marcus, Sep 12 2016

sumnumrat(1/(3*n-2)/n,1) \\ Charles R Greathouse IV, Feb 08 2023

Equals Sum_{n>=1} 1/(3*n^2 - 2*n).

Equals Pi/(4*sqrt(3)) + 3*log(3)/4. - Vaclav Kotesovec, Jul 05 2014

A244646 Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.2433209261537129892066077396310142821358441010300996244152817525...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Ravi P. Agarwal, Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series, Journal of Applied Mathematics and Physics, Vol.9, No.8 (2021), pp. 2038-2113. See p. 2076.
Wikipedia, Polygonal number.

Cf. A000038, A001106, A013661, A001620, A244639, A244644, A244645, A244647, A244648, A244649.

RealDigits[ Sum[2/(7n^2 - 5n), {n, 1 , Infinity}], 10, 111][[1]]

Equals Sum_{n>=1} 2/(7n^2 - 5n).
Equals (2*log(14) + 4*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14)) + Pi*tan(3*Pi/14))/5. - Vaclav Kotesovec, Jul 04 2014
Equals 14/25 - (2/5)*(gamma + psi(-5/7)), where gamma is Euler's constant (A001620) and psi(x) is the digamma function (Agarwal, 2021), psi(-5/7) = psi(2/7)+7/5 = -2.285517..., see A354628. - Amiram Eldar, Nov 12 2021

A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

For the partial sums of the reciprocals of the (positive) decagonal numbers see A250551(n+1)/A294515(n), n >= 0. - Wolfdieter Lang, Nov 07 2017

			1.216745956158244182494339352004760382108361700922772890949837441544696356350....

Wikipedia, Polygonal number

Cf. A001107, A000038, A013661, A244639, A016627, A244645, A244646, A244648, A244649, A250551(n+1)/A294515(n).

RealDigits[ Log[2] + Pi/6, 10, 111][[1]] (* or *)
RealDigits[ Sum[1/(4n^2 - 3n), {n, 1 , Infinity}], 10, 111][[1]]

log(2)+Pi/6 \\ Charles R Greathouse IV, Feb 08 2023

Sum_{n>0} 1/(4n^2 - 3n) = log(2) + Pi/6, (A002162 + A019673).

A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

From Wolfdieter Lang, Nov 09 2017: (Start)
In the Downey et al. link this is the instance k = 5 of the formula given there for S_{2*k+2}. A simpler formula is given in the Koecher reference as (5/4)*v_5(1) on p. 192. See the Kotesovec formula given below.
The partial sums are given in A294520/A294521. (End)

			1.1779560579226638587351739680918874184458572345666798028425228573...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Wikipedia, Polygonal number

Cf. A000038, A013661, A051624, A244639, A244644, A244645, A244646, A244647, A244648, A294520/A294521.

RealDigits[ Sum[1/(5n^2 - 4n), {n, 1 , Infinity}], 10, 111][[1]]

Equals Sum_{n>=1} 1/(5n^2 - 4n).
Equals Pi/8*sqrt(1+2/sqrt(5)) + (5*log(5) + sqrt(5)*log((3+sqrt(5))/2))/16. - Vaclav Kotesovec, Jul 04 2014
This is the value given in  the Koecher reference (see a comment above), and rewritten with the golden section phi = (1 + sqrt(5))/2 this becomes
  ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8. - Wolfdieter Lang, Nov 09 2017

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

Original entry on oeis.org

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

A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Sep 08 2023

This sequence is also the decimal expansion of Sum_{k>=1} 1/(f(k) +g(k)), where f(k) and g(k) are respectively the k-th triangular and the 13-gonal numbers (A000217 and A051865).

			0.63870452877981836559747674605121660577831724019512...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Ian Shamos, A catalog of the real numbers (2011), p. 544.
Wikipedia, Polygonal number.

Cf. A000217, A000796, A002194, A002391, A051865.

Cf. A244639, A244641, A244645, A244646, A244647, A244648, A244649.

SetDefaultRealField(RealField(139)); R:= RealField(); (Pi(R)*Sqrt(3)+9*Log(3))/24; // G. C. Greubel, Mar 24 2024

RealDigits[(Pi*Sqrt[3] + 9*Log[3])/24, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
(Pi*sqrt(3)+9*log(3))/24
```

numerical_approx((pi*sqrt(3)+9*log(3))/24, digits=139) # G. C. Greubel, Mar 24 2024

Equals Sum_{k>=1} 1/(6*k^2 - 4*k) = A244645/2 [Shamos].

Equals - Integral_{x=0..1} log(1-x^6)/x^5 dx [Shamos].

cp's OEIS Frontend

A244645 Decimal expansion of the sum of the reciprocals of the octagonal numbers (A000567).

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A244646 Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).

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A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

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A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

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

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A365522 Decimal expansion of (Pi*sqrt(3) + 9*log(3))/24.

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A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.