A000038 -id:A000038 - cp's OEIS frontend

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

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

A186501 Decimal expansion of the solution x to x^x = 11.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 22 2011

			2.5556046121008206152514542654716688251666246..

Cf. A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186502, A186503, A186504, A344930, A093590.

x=11;RealDigits[Log[x]/ProductLog[Log[x]],10,103][[1]]
RealDigits[x/.FindRoot[x^x==11,{x,2},WorkingPrecision->110]][[1]] (* Harvey P. Dale, Nov 13 2011 *)

A186502 Decimal expansion of the solution x to x^x = 12.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 22 2011

			2.6002950000539155877172082219411699164378377..

Cf. A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186503, A186504, A344930, A093590.

x=12;RealDigits[Log[x]/ProductLog[Log[x]],10,103][[1]]

A186503 Decimal expansion of the solution x to x^x = 13.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 22 2011

			2.6410619164843958084118390040657912549308732...

Cf. A016636, A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186502, A186504, A344930, A093590.

x=13;RealDigits[Log[x]/ProductLog[Log[x]],10,103][[1]]
RealDigits[x/.FindRoot[x^x==13,{x,2.6},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Feb 07 2025 *)

Equals log(13)/LambertW(log(13)). - Alois P. Heinz, Jun 16 2021

A186504 Decimal expansion of the solution x to x^x = 14.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 22 2011

			2.6785234858912995813011990010099506157786992..

Cf. A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186502, A186503, A344930, A093590.

x=14;RealDigits[Log[x]/ProductLog[Log[x]],10,200][[1]]
RealDigits[x/.FindRoot[x^x==14,{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 08 2023 *)

A344930 Decimal expansion of the real solution to x^x = 15.

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Jun 02 2021

2.7131636040042392095764012768285093718781...

Cf. A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186502, A186503, A186504, A093590.

RealDigits[Log[15]/ProductLog[Log[15]], 10, 100][[1]] (* Amiram Eldar, Jun 02 2021 *)
RealDigits[x/.FindRoot[x^x==15,{x,2.7},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Apr 18 2022 *)

Equals log(15)/W(log(15)), where W(z) is the Lambert W Function.

A364895 Decimal expansion of the 4-volume of the unit regular pentachoron (5-cell).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Aug 12 2023

Decimal expansion of sqrt(5)/96.

In general, the n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)).

			Equals 0.02329237476562280933...

Wikipedia, 5-cell
Polytope Wiki, Pentachoron

Decimal expansion of 4-volumes: this sequence (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), A364896 (120-cell), A364897 (600-cell).

Decimal expansion of the n-volume of the unit regular n-simplex: A120011 (n=2), A020829 (n=3), this sequence (n=4).

First[RealDigits[Sqrt[5]/96, 10, 100, -1]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
sqrt(5)/96
```

A061897 Square table by antidiagonals of number of routes of length 2k+n on the sides of a 2n-gon from a point to its opposite point.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 0, 4, 2, 0, 0, 8, 6, 2, 0, 0, 16, 18, 8, 2, 0, 0, 32, 54, 28, 10, 2, 0, 0, 64, 162, 96, 40, 12, 2, 0, 0, 128, 486, 328, 150, 54, 14, 2, 0, 0, 256, 1458, 1120, 550, 220, 70, 16, 2, 0, 0, 512, 4374, 3824, 2000, 858, 308, 88, 18, 2, 0, 0, 1024, 13122, 13056

Offset: 0

Henry Bottomley, May 14 2001

			Rows start
  1, 0, 0, 0, 0, ...
  2, 0, 0, 0, 0, ...
  2, 4, 8, 16, 32, ...
  2, 6, 18, 54, 162, ...
  2, 8, 28, 96, 328, ...
  ...

Cf. A060995. Rows include A000007, A000038, A000079, A008776, A060995. Columns effectively (i.e. except for a small number of terms) include A040000, A005843, A028552.

T(0, 0)=1; if n>0, T(n, 0)=2; if k>1, T(n, k)=T(n, k-1)*A061896(n, 1)-T(n, k-2)*A061896(n, 2)+T(n, k-3)*A061896(n, 3)-T(n, k-4)*A061896(n, 4)+...T(n, k-[n/2])*A061896(n, [n/2]); if 0>k T(n, k)=0.

A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 4, 0, 0, 1, 1, 4, 3, 1, 0, 1, 1, 6, 0, 0, 0, 0, 1, 1, 6, 2, 1, 1, 0, 0, 1, 1, 6, 2, 2, 0, 0, 0, 0, 1, 1, 8, 4, 0, 1, 1, 0, 0, 0, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 1, 1, 1;
  4, 0, 0, 1, 1;
  4, 3, 1, 0, 1, 1;
  6, 0, 0, 0, 0, 1, 1;
  6, 2, 1, 1, 0, 0, 1, 1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000010 (alternating row sums), A053158 (row sums).
Cf. A000038, A019590, A054523, A054977.
Cf. A063524, A097806, A126246, A183918.

A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
[A129479(n,k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024

A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[kA054523[n, j+k], {j,0,1}], 1];
Table[T[n,k],{n,16},{k,n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)

def A054523(n,k):
    if (k==n): return 1
    elif (n%k): return 0
    else: return euler_phi(n//k)
def A129479(n, k):
    if k<0 or k>n: return 0
    elif k==n: return 1
    else: return A054523(n,k) + A054523(n,k+1)
flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, Feb 11 2024

Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
T(n, 1) = A126246(n).
From G. C. Greubel, Feb 11 2024: (Start)
T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
T(2*n-1, n) = A019590(n).
T(2*n, n) = A054977(n).
T(2*n+1, n) = A000038(n).
T(3*n, n) = A063524(n-1).
T(3*n-2, n) = A183918(n+2).
Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)

cp's OEIS Frontend

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

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

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A186501 Decimal expansion of the solution x to x^x = 11.

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A186502 Decimal expansion of the solution x to x^x = 12.

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A186503 Decimal expansion of the solution x to x^x = 13.

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A186504 Decimal expansion of the solution x to x^x = 14.

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A344930 Decimal expansion of the real solution to x^x = 15.

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A364895 Decimal expansion of the 4-volume of the unit regular pentachoron (5-cell).

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PARI

A061897 Square table by antidiagonals of number of routes of length 2k+n on the sides of a 2n-gon from a point to its opposite point.

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