A275486 -id:A275486 - cp's OEIS frontend

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

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

Offset: 1

Bruno Berselli, Mar 06 2015

Value of the Borwein-Borwein function I_3(a,b) for a = b = 1. - Stanislav Sykora, Apr 16 2015

The area of a circle circumscribing a unit-area regular hexagon. - Amiram Eldar, Nov 05 2020

			1.2091995761561452337293855050947704881893774987284937170465899569254...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), pp. 120-121.
L. B. W. Jolley, Summation of Series, Dover (1961), No. 261, pp. 48, 49, (and No. 275).

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).
Su Hu and Min-Soo Kim, A generalization of Wallis' formula, arXiv:2201.09674 [math.NT], 2022.
Richard Kershner, The Number of Circles Covering a Set, American Journal of Mathematics, 61(3), 665-671.
MIT Integration Bee, 2023 MIT Integration Bee - Finals, Problem 1.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), C6.2.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See (9) p. 146.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean, equations 26-32.
Index entries for transcendental numbers

Cf. A000796, A001651, A002194, A093602, A248181, A257096, A257097, A186706.
Cf. A010815, A073010, A086089, A214549, A256923, A275486.
Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).

RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]

a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015

Equals 2*sqrt(3)*Pi/9 = 1 + 1/6 + 1/30 + 1/140 + 1/630 + 1/2772 + 1/12012 + ...
Equals m*I_3(m,m) = m*Integral_{x>=0} (x/(m^3+x^3)), for any m>0. - Stanislav Sykora, Apr 16 2015
Equals Integral_{x>=0} (1/(1+x^3)) dx. - Robert FERREOL, Dec 23 2016
From Peter Bala, Oct 27 2019: (Start)
Equals 3/4*Sum_{n >= 0} (n+1)!*(n+2)!/(2*n+3)!.
Equals Sum_{n >= 1} 3^(n-1)/(n*binomial(2*n,n)).
Equals 2*Sum_{n >= 1} 1/(n*binomial(2*n,n)). See Boros and Moll, pp. 120-121.
Equals Integral_{x = 0..1} 1/(1 - x^3)^(1/3) dx = Sum_{n >= 0} (-1)^n*binomial(-1/3,n) /(3*n + 1).
Equals 2*Sum_{n >= 1} 1/((3*n-1)*(3*n-2)) = 2*(1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...) (added Oct 30 2019). (End)
Equals Product_{k>=1} 9*k^2/(9*k^2 - 1). - Amiram Eldar, Aug 04 2020
From Peter Bala, Dec 13 2021: (Start)
Equals (2/3)*A093602.
Conjecture: for k >= 0, 2*sqrt(3)*Pi/9 = (3/2)^k * k!*Sum_{n = -oo..oo} (-1)^n/ Product_{j = 0..k} (3*n + 3*j + 1). (End)
Equals (3/4)*S - 1, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..Pi/2} tan(x)^(1/3)/(sin(2*x) + 1) dx. See MIT Link. - Joost de Winter, Aug 26 2023
Continued fraction: 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). See A000407. - Peter Bala, Feb 20 2024
Equals Sum_{n>=2} 1/binomial(n, floor(n/2)); and trivially if "floor" is replaced by "ceiling". - Richard R. Forberg, Aug 30 2024
Equals Product_{k>=2} (1 + (-1)^k/A001651(k)). - Amiram Eldar, Nov 22 2024
Equals 2*A073010 = 1/A086089 = sqrt(A214549) = exp(A256923) = A275486/2. - Hugo Pfoertner, Nov 22 2024
Equals 1 - (1/6) * Sum_{n>=1} A010815(n)/n. - Friedjof Tellkamp, Apr 05 2025
Equals A248181 - 2. - Pontus von Brömssen, Apr 05 2025

A113062 Expansion of theta series of hexagonal net with respect to a node.

Original entry on oeis.org

1, 3, 0, 6, 3, 0, 0, 6, 0, 6, 0, 0, 6, 6, 0, 0, 3, 0, 0, 6, 0, 12, 0, 0, 0, 3, 0, 6, 6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, 12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, 0, 6, 0, 12, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 6, 6, 0, 0, 6, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 0

Offset: 0

Michael Somos, Oct 13 2005

The hexagonal net is the familiar 2-dimensional honeycomb (not a lattice) in which each node has 3 neighbors.

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 3*q + 6*q^3 + 3*q^4 + 6*q^7 + 6*q^9 + 6*q^12 + 6*q^13 + 3*q^16 + ...

A. F. Wells, Structural Inorganic Chemistry, Oxford, 5th ed., 1984; see Fig. 3.9(a.1).

T. D. Noe, Table of n, a(n) for n = 0..1000
George L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From _N. J. A. Sloane_, Dec 18 2012
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A004016, A005882, A005928, A033685, A113063, A275486.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[0] = 1; a[n_] := 3*DivisorSum[n, {0, 1, -1, 1, 1, -1, -1, 1, -1}[[Mod[#, 9]+1]]&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 04 2015, after 1st PARI script *)

{a(n) = if( n<1, n==0, 3 * sumdiv(n, d, [ 0, 1, -1, 1, 1, -1, -1, 1, -1][d%9+1]))};

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 3 * prod(k=1, matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==3, 2, if(p%6==1, e+1, !(e%2))))))};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); abs( polcoeff( eta(x + A)^3 / eta(x^3 + A), n)))};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 6 * x * eta(x^9 + A)^3) / eta(x^3 + A), n))}; /* Michael Somos, Aug 15 2006 */

Moebius transform is period 9 sequence [ 3, -3, 3, 3, -3, -3, 3, -3, 0, ...].
Expansion of a(q^3) + c(q^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Aug 15 2006
For n>0, a(n) = 3*b(n) where b(n)=A113063(n) is multiplicative and b(p^e) = 2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
a(3*n + 2) = 0. a(3*n + 1) = A005882(n) = A033685(3*n + 1) = -A005928(3*n + 1). a(3*n) = A004016(n) = A005928(3*n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Dec 28 2023

Definition corrected Michael Somos, Oct 17 2005

A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 26 2006

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A097195, A107760, A112604, A112605, A113973, A121361, A123331, A123884, A275486.

Cf. A000700, A000122, A010054, A121373.

QP = QPochhammer; s = QP[q^2]*(QP[q^3]^6/(QP[q]^2*QP[q^6]^3)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, -(-1)^d * kronecker( -3, d)))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3), n))}

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 2*A[1] # Michael Somos, Sep 27 2013

Expansion of c(q)^2 / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
Expansion of phi(-x^3)^3 / phi(-x) where phi() is a Ramanujan theta function.
a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = (1 - 3*(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
Euler transform of period 6 sequence [ 2, 1, -4, 1, 2, -2, ...].
Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].
a(n) = 2 * A123331(n) if n>0. (-1)^n * a(n) = A113973(n).
G.f.: Product_{k>0} (1 + x^k)/(1 - x^k) * ((1 - x^(3*k)) / (1 + x^(3*k)))^3.
G.f.: 1 + 2 * Sum_{k>0} x^k / (1 - x^k + x^(2*k)) = theta_3(-x^3)^3 / theta_3(-x).
From Michael Somos, Aug 11 2009: (Start)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107760.
a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0.
a(2*n + 1) = 2 * A033762(n). a(3*n + 1) = 2 * A033687(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 3) = 2 * A112605(n). a(6*n + 1) = 2 * A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 4 * A121361(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Nov 14 2023

A217219 Theta series of planar hexagonal net (honeycomb) with respect to deep hole.

Original entry on oeis.org

0, 6, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A033685, A045833, A275486.

terms = 105; s = 6 q QPochhammer[q^9]^3/QPochhammer[q^3] + O[q]^(terms+5); CoefficientList[s, q][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017, after Michael Somos *)
CoefficientList[Series[6 q QPochhammer[q^9]^3/QPochhammer[q^3], {q, 0, 100}], q] (* G. C. Greubel, Aug 10 2018 *)

my(q='q+O('q^100)); concat([0], Vec(6*q*eta(q^9)^3/eta(q^3))) \\ G. C. Greubel, Aug 10 2018

See A033685, A045833.
a(n) = 2*A033685(n) = 6*A045833(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Oct 13 2022

Name edited by Andrey Zabolotskiy, Jun 21 2022

A373506 Decimal expansion of 4*Pi/3^(3/2) - Pi^2/9.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 07 2024

			1.321776441080139509810494232425524183566...

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

Cf. A100044, A275486, A073016 (no n+1 denominator), A073010 (denominator n), A373507 (denominator n-1).

Maple
```
4*Pi/3^(3/2)-Pi^2/9 ; evalf(%) ;
```

RealDigits[4*Pi/3^(3/2) - Pi^2/9, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

4*Pi/3^(3/2) - Pi^2/9 \\ Amiram Eldar, Jun 10 2024

Equals Sum_{n>=0} 1/((n+1)*binomial(2n,n)).
The alternating case is Sum_{n>=0} (-1)^n/((n+1)*binomial(2*n,n)) = 8*log(phi)/sqrt(5)-4*log^2(phi) = 0.79537... where phi is the golden ratio.
Equals A275486 - A100044. - Stefano Spezia, Jun 07 2024

A377008 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)(2/3)^(2*k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 12 2024

			0.88310581396712625588502373888562329082705924490169...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 272, eq. (29).

Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann zeta function.

Cf. A073005, A203129, A256923, A275486.

RealDigits[Log[4*Pi/(3*Sqrt[3])], 10, 120][[1]]
(* or *)
RealDigits[Log[Gamma[1/3]*Gamma[5/3]], 10, 120][[1]]

PARI
```
log(4*Pi/(3*sqrt(3)))
```
PARI
```
log(gamma(1/3)*gamma(5/3))
```

Equals log(4*Pi/(3*sqrt(3))) = log(A275486).

Equals log(Gamma(1/3)*Gamma(5/3)).

cp's OEIS Frontend

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

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A113062 Expansion of theta series of hexagonal net with respect to a node.

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A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

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A217219 Theta series of planar hexagonal net (honeycomb) with respect to deep hole.

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A373506 Decimal expansion of 4*Pi/3^(3/2) - Pi^2/9.

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A377008 Decimal expansion of Sum_{k>=1} (zeta(2*k)/k)*(2/3)^(2*k).

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A377008 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)(2/3)^(2*k).