A244649 -id:A244649 - 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).

A244648 Decimal expansion of the sum of the reciprocals of the hendecagonal numbers (A051682).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.195434116529627974352499234698499354884682627084658062386021603017...

Wikipedia, Polygonal number

Cf. A000038, A013661, A244639, A244644, A244645, A244646, A244647, A244649.

RealDigits[ Sum[2/(9n^2 - 7n), {n, 1 , Infinity}], 10, 111][[1]]

Sum_{n=1..infinity} 2/(9n^2 - 7n).

Equals (5*log(3) + Pi*cot(2*Pi/9) - 4*cos(2*Pi/9)*log(cos(Pi/18)) + 4*cos(Pi/9)*log(sin(2*Pi/9)) - 4*log(sin(Pi/9))*sin(Pi/18))/7. - Vaclav Kotesovec, Jul 04 2014

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

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 1) = A051624(k+1), for k >= 0.

Original entry on oeis.org

1, 13, 49, 795, 84179, 366829, 11417459, 103067441, 4235695001, 97604192047, 1661825059679, 1663957022369, 101611584435869, 101706166053389, 7226964017429851, 17176158550059533, 154681745346189277, 6654999228519884521, 6658297729691103841, 21316057915886595965, 2153790894613123442641

Offset: 0

Wolfdieter Lang, Nov 15 2017

The corresponding denominators are given in A294521.
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] = [5,1].
The limit of the series is V(5,1) = lim_{n -> oo} V(5,1;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8, with the golden section phi:= (1 + sqrt(5))/2. The value is 1.17795605792266... given in A244649.

			The rationals V(5,1;n), n >= 0, begin: 1, 13/12, 49/44, 795/704, 84179/73920, 366829/320320, 11417459/9929920, 103067441/89369280, 4235695001/3664140480, 97604192047/84275231040, 1661825059679/1432678927680, ...
V(5,1;10^6) = 1.177956058 (Maple, 10 digits) to be compared with 1.177956058 obtained from V(5,1) given in A244649.

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

G. C. Greubel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A051624, A244649, A294512, A294516/A294517, A294521.

[Numerator((&+[1/((k+1)*(5*k+1)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

Table[Numerator[Sum[1/((k + 1)*(5*k + 1)), {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Aug 29 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 1)))); \\ Michel Marcus, Nov 15 2017

a(n) = numerator(V(5,1;n)) with V(5,1;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 1)) = Sum_{k=0..n} 1/A051624(k+1) = (1/4)*Sum_{k=0..n} (1/(k + 1/5) - 1/(k+1)) = (-Psi(1/5) + Psi(n+6/5) - (gamma + Psi(n+2)))/4, with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294826 Numerators of the partial sums of the reciprocals of twice the heptagonal numbers (k + 1)(5k + 2) = A135706(k+1) = 2*A000566(k+1), for k >= 0.

Original entry on oeis.org

1, 4, 151, 1315, 36698, 667109, 10749479, 399851303, 401511863, 18933826729, 246810236317, 4700047812703, 145981746528913, 9796912235587651, 9810925971351679, 9823210739716249, 403196782523223569, 11704197956499986461, 269433333504358946963, 5231145593209503407215, 747842028258712790473

Offset: 0

Wolfdieter Lang, Nov 16 2017

The corresponding denominators are given in A294827.
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] = [5,2].
The limit of the series is V(5,2) = lim_{n -> oo} V(5,2;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) - (Pi/5)*sqrt(7-4*phi)))/6, with the golden section phi:= (1 + sqrt(5))/2. The value is 0.661389626561... given by (1/2)*A244639.
In the Koecher reference v_5(2) =  (3/5)*V(5,2) = 0.39683377593671665701 ...is given as (1/4)*log(5) - (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) + (Pi/10)*sqrt((5 - 2*sqrt(5))/5).

			The rationals V(5,2;n), n >= 0, begin: 1/2, 4/7, 151/252, 1315/2142, 36698/58905, 667109/1060290, 10749479/16964640, 399851303/627691680, 401511863/627691680, 18933826729/29501508960, 246810236317/383519616480, ...
V(5,2;10^6) = 0.6613894266 (Maple, 10 digits) to be compared with 0.6613896266 giving the 10 digit value of V(5,2) from (1/2)*A244649.

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

G. C. Greubel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A000566, A135706, A294512, A294520/A294521 (V(5,1;n)), A244639.

[Numerator((&+[1/((k+1)*(5*k+2)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

Table[Numerator[Sum[1/((k+1)*(5*k+2)), {k,0,n}]], {n,0,25}] (* G. C. Greubel, Aug 29 2018 *)
Accumulate[1/(2*PolygonalNumber[7,Range[30]])]//Numerator (* Harvey P. Dale, Aug 31 2023 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 2)))); \\ Michel Marcus, Nov 17 2017

a(n) = numerator(V(5,2;n)) with V(5,2;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 2)) = Sum_{k=0..n} 1/A135706(k+1) = (1/3)*Sum_{k=0..n} (1/(k + 2/5) - 1/(k+1)) = (-Psi(2/5) + Psi(n+7/5) - (gamma + Psi(n+2)))/3 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A244648 Decimal expansion of the sum of the reciprocals of the hendecagonal numbers (A051682).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)*(5*k + 1) = A051624(k+1), for k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A294826 Numerators of the partial sums of the reciprocals of twice the heptagonal numbers (k + 1)*(5*k + 2) = A135706(k+1) = 2*A000566(k+1), for k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 1) = A051624(k+1), for k >= 0.

A294826 Numerators of the partial sums of the reciprocals of twice the heptagonal numbers (k + 1)(5k + 2) = A135706(k+1) = 2*A000566(k+1), for k >= 0.

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