A073010 -id:A073010 - cp's OEIS frontend

A113973 Expansion of phi(x^3)^3/phi(x) where phi() is a Ramanujan theta function.

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, Nov 10 2005

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1985, see p. 375, Entry 35.

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.

a(n)=-2*A113974(n) if n>0.

Cf. A000700, A000122, A010054, A073010, A121373.

s = EllipticTheta[3, 0, q^3]^3/EllipticTheta[3, 0, q] + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 04 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := ((-1)^e - 3)/2; f[3, e_] := 1; a[0] = 1; a[1] = -2; a[n_] := -2 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Nov 14 2023 *)

{a(n)=local(x); if(n<1, n==0, x=valuation(n,2); if(n%2,-2,(3-(-1)^x))*sumdiv(n/2^x,d, kronecker(-3,d)))}

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

{a(n)=if(n<1, n==0, -2*direuler(p=2,n, if(p==2, 2-(1+2*X)/(1-X^2), 1/(1-X)/(1-kronecker(-3,p)*X)))[n])}

{a(n)=local(A); if(n<0, 0, A=sum(k=1,sqrtint(n), 2*x^k^2, 1+x*O(x^n)); polcoeff( subst(A+x*O(x^(n\3)),x,x^3)^3/A, n))}

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)^2*eta(x^6+A)^15/ eta(x^2+A)^5/eta(x^3+A)^6/eta(x^12+A)^6, n))}

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 12 sequence [ -2, 3, 4, 1, -2, -6, -2, 1, 4, 3, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 6, 0, -2, 2, 0, -2, 2, 0, -6, 2, 0, ...].
Expansion of (eta(q)^2*eta(q^4)^2*eta(q^6)^15)/ (eta(q^2)^5*eta(q^3)^6*eta(q^12)^6) in powers of q.
G.f.: theta_3(q^3)^3/theta_3(q).
G.f.: 1+2( Sum_{k>0} x^(3k-1)/(1-(-x)^(3k-1)) - x^(3k-2)/(1-(-x)^(3k-2))) = 1 +2( Sum_{k>0} (-1)^k x^k/(1+x^k+x^(2k)) +2 x^(4k)/(1+x^(4k)+x^(8k)) ).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 14 2023

A113476 Decimal expansion of (log(2) + Pi/sqrt(3))/3.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 08 2006

This number is transcendental - this follows from a result of Baker (1968) on linear forms of algebraic numbers.

			0.835648848264721053337... = A073010 + A193535.

Jolley, Summation of Series, Dover (1961), eq (79) page 16.
Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equation 21.16

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. Baker, Linear forms in the logarithms of algebraic numbers (IV). Mathematika, 15 (1968) pp. 204-216
L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 5
Eric W. Weisstein, Euler's Series Transformation.
Index entries for transcendental numbers

Cf. A007559, A016777, A024217, A073010, A193534, A193535.

RealDigits[(Log[2]+\[Pi]/Sqrt[3])/3,10,120][[1]]  (* Harvey P. Dale, Mar 26 2011 *)

PARI
```
1/3*(log(2)+Pi/sqrt(3))
```

Equals Integral_{x = 0..1} dx/(1+x^3) = Sum_{k >= 0} (-1)^k/(3*k+1) = 1 - 1/4 + 1/7 - 1/10 + 1/13 - 1/16 + ... (see A016777). - Benoit Cloitre, Alonso del Arte, Jul 29 2011
Generalized continued fraction: 1/(1 + 1^2/(3 + 4^2/(3 + 7^2/(3 + 10^2/(3 + ... ))))) due to Euler. For a sketch proof see A024217. - Peter Bala, Feb 22 2015
Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 1) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A007559(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 1)). - Peter Bala, Dec 01 2021
From Peter Bala, Mar 03 2024: (Start)
Equals hypergeom([1/3,  1], [4/3], -1).
Gauss's continued fraction: 1/(1 + 1/(4 + 3^2/(7 + 4^2/(10 + 6^2/(13 + 7^2/(16 + 9^2/(19 + 10^2/(22 + 12^2/(25 + 13^2/(28 + ... )))))))))). (End)
Equals (1/12) * Sum_{n >= 0} (-1/2)^n * (9*n + 7)/((3*n + 2)*(n + 1)*binomial(2*n+1/3, n+1)). - Peter Bala, Mar 05 2025

A193534 Decimal expansion of (1/3) * (Pi/sqrt(3) - log(2)).

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Jul 29 2011

The formulas for this number and the constant in A113476 are exactly the same except for one small, crucial detail: the infinite sum has a denominator of 3i + 2 rather than 3i + 1, while in the closed form, log(2)/3 is subtracted from rather than added to (Pi * sqrt(3))/9.

Understandably, the typesetter for Spiegel et al. (2009) set the closed formula for this number incorrectly (as being the same as for A113476, compare equation 21.16 on the same page of that book).

			0.373550727891424180392282045394659721402855371244161773816401641964909853052219...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (80), page 16.
J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 132.
Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill, 2009, p. 135, equation 21.18.

Gheorghe Coserea, Table of n, a(n) for n = 0..2015
Eric W. Weisstein, Euler's Series Transformation.

Cf. A073010, A193535, A024396, A113476, A196548, A258969.

evalf((Psi(5/6)-Psi(1/3))/6, 120); # Vaclav Kotesovec, Jun 16 2015

RealDigits[(Pi Sqrt[3])/9 - (Log[2]/3), 10, 100][[1]]

(Pi/sqrt(3)-log(2))/3 \\ Charles R Greathouse IV, Jul 29 2011

default(realprecision, 98);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(3*n+2)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015

Equals Sum_{k >= 0} (-1)^k/(3k + 2) = 1/2 - 1/5 + 1/8 - 1/11 + 1/14 - 1/17 + ... (see A016789).
From Peter Bala, Feb 20 2015: (Start)
Equals (1/2) * Integral_{x = 0..1} 1/(1 + x^(3/2)) dx.
Generalized continued fraction: 1/(2 + 2^2/(3 + 5^2/(3 + 8^2/(3 + 11^2/(3 + ... ))))) due to Euler. For a sketch proof see A024396. (End)
Equals (Psi(5/6)-Psi(1/3))/6. - Vaclav Kotesovec, Jun 16 2015
Equals Integral_{x = 1..infinity} 1/(1 + x^3) dx. - Robert FERREOL, Dec 23 2016
Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 2) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A008544(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 2)). - Peter Bala, Dec 01 2021
From Bernard Schott, Jan 28 2022: (Start)
Equals Integral_{x = 0..1} x/(1+ x^3) dx (see Rivaud reference).
Equals 3 * A196548. (End)
From Peter Bala, Mar 03 2024: (Start)
Equals (1/2)*hypergeom([2/3, 1], [5/3], -1).
Gauss's continued fraction: 1/(2 + 2^2/(5 + 3^2/(8 + 5^2/(11 + 6^2/(14 + 8^2/(17 + 9^2/(20 + 11^2/(23 + 12^2/(26 + ... ))))))))). (End)

A001504 a(n) = (3n+1)(3*n+2).

Original entry on oeis.org

2, 20, 56, 110, 182, 272, 380, 506, 650, 812, 992, 1190, 1406, 1640, 1892, 2162, 2450, 2756, 3080, 3422, 3782, 4160, 4556, 4970, 5402, 5852, 6320, 6806, 7310, 7832, 8372, 8930, 9506, 10100, 10712, 11342, 11990, 12656, 13340, 14042, 14762, 15500, 16256, 17030

Offset: 0

N. J. A. Sloane

The oblong numbers (A002378) not divisible by 3. - Gionata Neri, May 10 2015

The continued fraction expansion of sqrt(a(n)+1) is [3n+1; {1, 1, 2n, 1, 1,6n+2}]. For n=0, this collapses to [1; {1, 2}]. - Magus K. Chu, Nov 13 2024

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Subsequence of A002378.

Cf. A016777, A016789, A060544, A073010, A387235.

A001504:=n->(3*n+1)*(3*n+2): seq(A001504(n), n=0..100); # Wesley Ivan Hurt, Jan 29 2017

Table[(3*n+1)*(3*n+2),{n,50}] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)
LinearRecurrence[{3,-3,1},{2,20,56},80] (* Harvey P. Dale, Mar 16 2025 *)

a(n)=(3*n+1)*(3*n+2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A060544(n+1)*2.
Sum_{k>=0} 1/a(k) = (Pi/3)/sqrt(3) = A073010. - Benoit Cloitre, Aug 20 2002
a(n) = 18*n + a(n-1) with a(0) = 2. - Vincenzo Librandi, Nov 12 2010
Sum_{n>=0} (-1)^n/a(n) = 2*log(2)/3 (A387235). - Amiram Eldar, Jan 14 2021
G.f.: -2*(x^2+7*x+1)/(x-1)^3. - Alois P. Heinz, Feb 28 2021
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A016777(n)*A016789(n).
Product_{n>=0} (1 - 1/a(n)) = 2*cos(sqrt(5)*Pi/6)/sqrt(3).
Product_{n>=0} (1 + 1/a(n)) = 2*cosh(sqrt(3)*Pi/6)/sqrt(3). (End)
E.g.f.: exp(x)*(2 + 18*x + 9*x^2). - Stefano Spezia, Aug 23 2025

A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 2, 1, 1, 0, 0, -1, 2, 2, 0, -1, 0, 1, 2, 0, 2, 0, 0, 1, 1, 2, 1, -2, 0, 0, 2, 1, 0, 0, 0, -1, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, -1, 3, 1, 0, -2, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, -1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 1, -2, 0, 2, 2, 0, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 3, 0, -1, 0, 0, 2, 2, 0

Offset: 1

Michael Somos, Nov 02 2005

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

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

			G.f. = q + q^2 + q^3 - q^4 + q^6 + 2*q^7 + q^8 + q^9 - q^12 + 2*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Li-Chien Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1105, equation (3.8).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A073010, A093829, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964. A133637.

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

A := Basis( ModularForms( Gamma1(24), 1), 106); A[2] + A[3] + A[4] - A[5] + A[7] + 2*A[8] + A[9] + A[10]; /* Michael Somos, May 07 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, Jan 31 2015 *)

{a(n) = if( n<1, 0, -(-1)^max( 1, valuation( n, 2)) * sumdiv(n, d, kronecker( -12, d)))};

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==2, 1 + X / (1 + X), 1 / ((1 - X) * (1 - kronecker( -12, p) * X))))[n])};

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3), n))};

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 0, -2, -1, 0, 1, 2, 0, 0,-1][d%12 + 1]))}; /* Michael Somos, May 07 2015 */

Expansion of (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = -(-1)^e if e>0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - 2 * x^(12*k - 8) / (1 - x^(12*k - 8)) + 2 * x^(12*k - 4) / (1 - x^(12*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(3*k))^2 / (1 + x^(4*k) + x^(8*k)).
G.f.: x * Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)) * ((1 - x^(12*k - 6)) / (1 - x^(3*k)))^3.
Expansion of theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
Expansion of q * psi(-q^3)^3 / psi(-q) in powers of q where psi() is a Ramanujan theta function.
Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132973.
a(n) = -(-1)^n * A093829(n). - Michael Somos, Jan 31 2015
Convolution inverse of A133637.
a(3*n) = a(n). a(6*n + 5) = a(12*n + 10) = 0. |a(n)| = A035178(n).
a(2*n) = A093829(n). a(2*n + 1) = A033762(n).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 6) = A112605(n). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A121963(n). a(24*n + 19) = 2 * A131964(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 23 2023

A136016 a(n) = 9*n^2-1.

Original entry on oeis.org

8, 35, 80, 143, 224, 323, 440, 575, 728, 899, 1088, 1295, 1520, 1763, 2024, 2303, 2600, 2915, 3248, 3599, 3968, 4355, 4760, 5183, 5624, 6083, 6560, 7055, 7568, 8099, 8648, 9215, 9800, 10403, 11024, 11663, 12320, 12995, 13688, 14399, 15128, 15875, 16640

Offset: 1

Artur Jasinski, Dec 10 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001359, A006512, A037074, A248897.

Cf. A005563, A016777, A016789, A073010, A136017.

[9*n^2-1: n in [1..50]]; // Vincenzo Librandi, May 09 2011

Table[9n^2 - 1, {n, 1, 100}]
LinearRecurrence[{3,-3,1},{8,35,80},50] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=9*n^2-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A005563(3*n-1). - Paul Curtz, Oct 28 2008
a(2*n) = A136017(n). - Paul Curtz, Sep 30 2008
a(n) = A016777(n)*A016789(n-1). - Reinhard Zumkeller, Feb 15 2009
G.f.: x*(-8-11*x+x^2) / ( x-1 )^3. - R. J. Mathar, Jul 01 2011
From Amiram Eldar, Jul 31 2020: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - sqrt(3)*Pi/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/9 - 1/2. (End)
From Amiram Eldar, Feb 04 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 2*Pi/(3*sqrt(3)) (A248897).
Product_{n>=1} (1 - 1/a(n)) = sqrt(2/3)*sin(sqrt(2)*Pi/3). (End)
a(n) = a(-n) for all n in Z. Sum_{n in Z} 1/a(n) = -Pi/3^(3/2) = -A073010. - Michael Somos, May 21 2023
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jun 19 2025

A240935 Decimal expansion of 3sqrt(3)/(4Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 03 2014

A triangle of maximal area inside a circle is necessarily an inscribed equilateral triangle. This constant is the ratio of the triangle's area to the circle's area. In general, the ratio of an arbitrary triangle's area to the area of its unique Steiner ellipse, which has the least area of any circumscribed ellipse (an equilateral triangle's Steiner ellipse is a circle).

Also the probability that the distance between 2 randomly selected points within a circle will be larger than the radius. - Amiram Eldar, Mar 03 2019

			0.4134966715663440371334948737347270810480...

B. F. Finkel, Problem 38, solved by O. W. Anthony, Henry Heaton, and G. B. M. Zerr, The American Mathematical Monthly, Vol. 3, No. 12 (1896), pp. 324-326.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p.52
I. Todhunter, A treatise on the Integral Calculus, London and Cambridge: MacMillan and Co., 1868, page 320, Example 7.
Wikipedia, Steiner ellipse
Index entries for transcendental numbers

Cf. A073010, A060294, A003881, A002194, A000796, A102519, A086089.

Digits:=100: evalf(3*sqrt(3)/(4*Pi)); # Wesley Ivan Hurt, Aug 03 2014

Flatten[RealDigits[3 Sqrt[3]/(4 Pi), 10, 100, -1]] (* Wesley Ivan Hurt, Aug 03 2014 *)

default(realprecision, 120);
3*sqrt(3)/(4*Pi)

3*sqrt(3)/(4*Pi) = 3*A002194/(4*A000796).

Equals A093604^2. - Hugo Pfoertner, May 18 2024

A381152 Decimal expansion of the isoperimetric quotient of a regular pentagon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

The isoperimetric quotient of a closed curve is equal to 4*Pi*A/p^2, where A is the area enclosed by the curve and p is its perimeter. For a regular n-gon, this is equivalent to Pi/(n*tan(Pi/n)).

The isoperimetric quotient of a circle is 1.

			0.86480626597720996723118206585862333703828555690228...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019952, A102771.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(5*Tan[Pi/5]), 10, 100]]

Equals Pi/(5*tan(Pi/5)) = (Pi/5)*A019952.

Equals (4/25)*Pi*A102771.

A381153 Decimal expansion of the isoperimetric quotient of a regular heptagon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.93194062349909574595222630089422754574528525154715...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A178817, A343058.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(7*Tan[Pi/7]), 10, 100]]

Equals Pi/(7*tan(Pi/7)) = Pi/(7*A343058).

Equals (4/49)*Pi*A178817.

A381154 Decimal expansion of the isoperimetric quotient of a regular 9-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.959050541873609358074543306708643413020181580975...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019918, A256853.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(9*Tan[Pi/9]), 10, 100]]

Equals Pi/(9*tan(Pi/9)) = Pi/(9*A019918).

Equals (4/81)*Pi*A256853.

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A113973 Expansion of phi(x^3)^3/phi(x) where phi() is a Ramanujan theta function.

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A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

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A136016 a(n) = 9*n^2-1.

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A001504 a(n) = (3n+1)(3*n+2).

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