A073010 -id:A073010 - cp's OEIS frontend

A261850 Decimal expansion of the central binomial sum S(6), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.50267652147826928645467745997934863966460260009164...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261851 (S(7)), A261852 (S(8)).

S[6] = Sum[1/(n^6*Binomial[2n, n]), {n, 1, Infinity}]; RealDigits[S[6], 10, 105]//First

Equals (1/2) 7F6(1,1,1,1,1,1,1; 3/2,2,2,2,2,2; 1/4).

Also equals (2/3)*Integral_{0..Pi/3} t*log(2*sin(t/2))^4 dt.

A261851 Decimal expansion of the central binomial sum S(7), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.501325872688178809402296710552749443726878329858...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261852 (S(8)).

S[7]=-6*Pi*Im[-PolyLog[6, (-1)^(1/3)]] + (17*Pi^4*Zeta[3])/1620 + (1/3)*Pi^2*Zeta[5] - (493*Zeta[7])/24; RealDigits[S[7], 10, 105]//First

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals -6*Pi*Im(-PolyLog(6, (-1)^(1/3))) + (17*Pi^4*zeta(3))/1620 + (1/3)*Pi^2*zeta(5) - (493*zeta(7))/24.

A261852 Decimal expansion of the central binomial sum S(8), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.5006588912976705433145571270829868383840732523404540388886438...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261851 (S(7)).

S[8] = Sum[1/(n^8*Binomial[2n, n]), {n, 1, Infinity}]; RealDigits[S[8], 10, 100] // First

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals (4/45)*Integral_{0..Pi/3} t*log(2*sin(t/2))^6 dt.

A263498 Decimal expansion of the Gaussian Hypergeometric Function 2F1(1, 3; 5/2; x) at x=1/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 19 2015

Division through 3 gives 0.472799.. = integral_{x=0..infinity} x^2*I_1(x)*K_1(x)^2 dx, where I and K are Modified Bessel Functions.

			1.41839915231229046745877101018954097637875499745698743409317991385...

Cf. A073010.

RealDigits[4*Pi/3^(3/2) - 1, 10, 120][[1]] (* Vaclav Kotesovec, Apr 10 2016 *)

4*Pi/sqrt(27)-1 \\ Charles R Greathouse IV, Aug 01 2016

Equals 4*Pi/3^(3/2) - 1. - Vaclav Kotesovec, Apr 10 2016

A273989 Decimal expansion of the odd Bessel moment t(5,1) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 06 2016

			0.660344869018672357837266831705994263854241991696873858300803587553894...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008, page 21.

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273985 (s(5,3)), A273986 (s(5,5)), A273990 (t(5,3)), A273991 (t(5,5)).

t[5, 1] = NIntegrate[x*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 105]; RealDigits[t[5, 1]][[1]]
(* or: *)
t[5, 1] = 4(7 - 4*Sqrt[3]) EllipticK[1 - 32/(16 + 7*Sqrt[3] - Sqrt[15])] EllipticK[1 - 32/(16 + 7*Sqrt[3] + Sqrt[15])]; RealDigits[t[5, 1], 10, 105][[1]]
RealDigits[EllipticK[(16 - 7 Sqrt[3] - Sqrt[15])/32] EllipticK[(16 - 7 Sqrt[3] + Sqrt[15])/32]/4, 10, 105][[1]] (* Jan Mangaldan, Jan 06 2017 *)

Integral_{0..inf} x*BesselI_0(x)^2*BesselK_0(x)^3.

Equals 4(7 - 4*sqrt(3)) EllipticK(1 - 32/(16 + 7*sqrt(3) - sqrt(15))) EllipticK(1 - 32/(16 + 7*sqrt(3) + sqrt(15))).

A273990 Decimal expansion of the odd Bessel moment t(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 06 2016

			0.252253944897841965994509612555090408775068450755970099920659309452897...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008, page 21.

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273985 (s(5,3)), A273986 (s(5,5)), A273989 (t(5,1)), A273991 (t(5,5)).

t[5, 3] = NIntegrate[x^3*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 104];
RealDigits[t[5, 3]][[1]]
(* or: *)
K[k_] := EllipticK[k^2/(-1+k^2)]/Sqrt[1-k^2];
D0[y_] := (4*y*K[Sqrt[((1-3*y)*(1+y)^3)/((1+3*y)*(1-y)^3)]])/Sqrt[(1+3*y)* (1-y)^3];
t[5, 3] = NIntegrate[4y^2*(1-2y^2+4y^4)*D0[y]/(1-4y^2)^(5/2), {y, 0, 1/3}, WorkingPrecision -> 104];
RealDigits[t[5, 3]][[1]]

Integral_{0..inf} x*BesselI_0(x)^2*BesselK_0(x)^3.

A273991 Decimal expansion of the odd Bessel moment t(5,5) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 06 2016

			1.0432973673868713449189316078947712217566163312269155788688325589866...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008, page 21.

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273985 (s(5,3)), A273986 (s(5,5)), A273989 (t(5,1)), A273990 (t(5,3)).

t[5, 5] = NIntegrate[x^5*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 103];
RealDigits[t[5, 5]][[1]]

Integral_{0..inf} x^5*BesselI_0(x)^2*BesselK_0(x)^3.

Conjecture: Equals 76/15 t(5,3) - 16/45 t(5,1).

A306712 Decimal expansion of 3*sqrt(3)/Pi.

Original entry on oeis.org

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

Offset: 1

Scott R. Shannon, Mar 05 2019

This is the mean end-to-end distance of the 2-step self-avoiding walk with full excluded volume in the 2-dimensional continuum.
Take 3 touching circles of diameter 1 which are joined as a chain and each is free to move around its neighbors' perimeters, but no circle can overlap another. This value is the average of the distance from the middle of the first circle to the middle of the third circle, averaged over all possible configurations the chain of 3 non-overlapping circles can take.
Using the law of cosines one can show the distance between the middle of the first and third circles, r_3, in the 3-circle chain is r_3 = sqrt(2-2*cos(t)), where t is the angle between these circles centered on the second circle. The mean end-to-end distance is thus given by the integral  = Integrate(r_3,{t,Pi/3,5*Pi/3})/(4*Pi/3), which includes division by the required normalization constant. Solving this definite integral gives the exact value for  as 3*sqrt(3)/Pi. This is A289504 minus 2.
Removing the square root from r_3 in the above integral gives the mean square end-to-end distance for the 2-step walk. Evaluating this integral gives the exact value for  as 2+3*sqrt(3)/(2*Pi), with a value of approximately 2.826993343... . This is A086089 plus 2, or equivalently this sequence divided by 2, plus 2.

			1.653986686265376148533979494938908324192159441099921958398...

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Java code for a 2-step self-avoiding walk in the continuum. This produces a numerical approximation to the exact values.
Index entries for transcendental numbers.

Cf. A306648, A086089, A289504, A073010 (reciprocal).

RealDigits[3*Sqrt[3]/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

PARI
```
3*sqrt(3)/Pi
```

Terms a(59) and beyond from Andrew Howroyd, Apr 27 2020

A373507 Decimal expansion of 1/2 - sqrt(3)*Pi/18.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 07 2024

			0.1977001059609636915676536237263...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.

Cf. A373508 (denominator (n-1)^2), A373506 (denominator n+1), A073010 (denominator n).

Maple
```
1/2-sqrt(3)*Pi/18; evalf(%) ;
```

RealDigits[1/2 - Sqrt[3]*Pi/18, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

1/2 - sqrt(3)*Pi/18 \\ Amiram Eldar, Jun 10 2024

Equals Sum_{n>=2} 1/((n-1)*binomial(2n,n)).

Sum_{n>=2} (-1)^n/((n-1)*binomial(2n,n)) = 3*log(phi)/sqrt(5) - 1/2 = 0.145613... where phi is the golden ratio.

A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

For the definition of isoperimetric quotient of a solid, see A381671.

			0.8287977192520120215005810038129635758617830308723...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273637 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381671 (tetrahedron).
Cf. A000796, A002194, A374883.

First[RealDigits[Pi*GoldenRatio^4/(15*Sqrt[3]), 10, 100]]

Equals Pi*phi^4/(15*sqrt(3)) = A000796*A374883/(15*A002194).

cp's OEIS Frontend

A261850 Decimal expansion of the central binomial sum S(6), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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A261851 Decimal expansion of the central binomial sum S(7), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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A261852 Decimal expansion of the central binomial sum S(8), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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A263498 Decimal expansion of the Gaussian Hypergeometric Function 2F1(1, 3; 5/2; x) at x=1/4.

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A273989 Decimal expansion of the odd Bessel moment t(5,1) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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A273990 Decimal expansion of the odd Bessel moment t(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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A273991 Decimal expansion of the odd Bessel moment t(5,5) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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A306712 Decimal expansion of 3*sqrt(3)/Pi.

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