A019669 -id:A019669 - cp's OEIS frontend

A228719 Decimal expansion of 3*Pi/5.

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

Offset: 1

Omar E. Pol, Oct 03 2013

With offset 2, decimal expansion of 6*Pi.
6*Pi is also the surface area of a sphere whose diameter equals the square root of 6. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013
The interior angle of a regular pentagon expressed in radians. - Stefano Spezia, May 30 2025

			3*Pi/5 = 1.8849555921538759430775860299677017305183...
6*Pi = 18.849555921538759430775860299677017305183...

Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669 (Pi through 5*Pi).

Cf. A374172.

First[RealDigits[6*Pi, 10, 100]] (* Paolo Xausa, Jul 16 2024 *)

 6*Pi \\ Charles R Greathouse IV, Sep 30 2022

Equals 1/(10*A374172). - Hugo Pfoertner, Jul 16 2024

A094642 Decimal expansion of log(Pi/2).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow and Robert G. Wilson v, May 18 2004

			log(Pi/2) = 0.45158270528945486472619522989488214357179467855505...

George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Dirk Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly, Vol. 108, No. 3 (2001), pp. 222-231.
Jonathan Sondow, A faster product for pi and a new integral for ln(pi/2), The American Mathematical Monthly, Vol. 112, No. 8 (2005), pp. 729-734; Editor's endnotes, ibid., Vol. 113, No. 7 (2006), pp. 670-671; arXiv preprint, arXiv:math/0401406 [math.NT], 2004.

Cf. A019669, A094643.

Mathematica
```
RealDigits[ Log[Pi/2], 10, 111][[1]]
```

log(Pi/2) \\ Charles R Greathouse IV, Jun 23 2014

Equals Sum_{n>=1} zeta(2*n)/(n*2^(2*n)) (cf. Boros & Moll p. 131). - Jean-François Alcover, Apr 29 2013
Equals Re(log(log(I))). - Stanislav Sykora, May 09 2015
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/cosh(Pi*z) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Integral_{0..Pi/2} (2/(Pi-2*t)-tan(t)) dt. - Clark Kimberling, Jul 10 2020
Equals -Sum_{k>=1} log(1 - 1/(2*k)^2). - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} (-1)^(k+1) * log(1 + 1/k). - Amiram Eldar, Jun 26 2021
Equals A053510 - A002162. - R. J. Mathar, Jun 15 2023

A137218 Decimal expansion of the argument of -1 + 2*i.

Original entry on oeis.org

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

Offset: 1

Matt Rieckman (mjr162006(AT)yahoo.com), Mar 06 2008

Gives closed forms for many arctangent values:
arctan(2) = Pi - a, arctan(1/2) = a - Pi/2,
arctan(3) = a - Pi/4, arctan(1/3) = 3*Pi/4 - a,
arctan(7) = 7*Pi/4 - 2*a, arctan(1/7) = 2*a - 5*Pi/4,
arctan(4/3) = 2*a - Pi and arctan(3/4) = 3*Pi/2 - 2*a.
Dihedral angle in the dodecahedron (radians). - R. J. Mathar, Mar 24 2012
Larger interior angle (in radians) of a golden rhombus; A105199 is the smaller interior angle. - Eric W. Weisstein, Dec 17 2018

			2.0344439357957027354455779231...

Rick L. Shepherd, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Dodecahedron
Index entries for transcendental numbers.

Platonic solids' dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A236367 (icosahedron). - Stanislav Sykora, Jan 23 2014
Cf. A242723 (same in degrees).
Cf. A105199 (smaller interior angle of the golden rhombus).

RealDigits[Pi - ArcTan[2], 10, 120][[1]] (* Harvey P. Dale, Aug 08 2014 *)

default(realprecision, 120);
acos(-1/sqrt(5)) \\ or
arg(-1+2*I) \\ Rick L. Shepherd, Jan 26 2014

Equals Pi - arctan(2) = A000796 - A105199 = 2*A195723.

Corrected a typo in the sequence Matt Rieckman (mjr162006(AT)yahoo.com), Feb 05 2010

More terms from Rick L. Shepherd, Jan 26 2014

A019683 Decimal expansion of Pi/16.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			Pi/16 = 0.19634954084936207740391521145496893026232308746094411381... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4.2, p. 494.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A157332, A000796, A003881, A019669, A019670, A019675, A019679, A244978.

R:= RealField(100); Pi(R)/16; // G. C. Greubel, Aug 26 2019

RealDigits[N[Pi/16,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 100); Pi/16 \\ G. C. Greubel, Aug 26 2019

numerical_approx(pi/16, digits=100) # G. C. Greubel, Aug 26 2019

From  Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^3*sqrt(1 - x^8) dx.
Equals Integral_{x = 0..inf} x^2/(1 + x^2)^3 dx. (End)
From Amiram Eldar, Aug 04 2020: (Start)
Equals Sum_{k>=1} sin(k)^3 * cos(k)/k.
Equals Sum_{k>=1} sin(k)^3 * cos(k)^2/k.
Equals Sum_{k>=1} (-1)^(k+1) * sin((2*k-1)/4)/(2*k-1)^2. (End)

A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 3, 0, 1, 1, 2, 0, 3, 2, 0, 6, 1, 2, 1, 0, 2, 0, 0, 0, 3, 3, 2, 3, 0, 2, 6, 0, 1, 0, 2, 0, 1, 2, 0, 6, 2, 2, 0, 0, 0, 2, 0, 0, 3, 1, 3, 6, 2, 2, 3, 0, 0, 0, 2, 0, 6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 9, 0, 0, 6, 0, 2, 1, 2, 0, 0, 4, 0, 6, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 1, 0, 3, 2, 6, 0, 2

Offset: 0

Michael Somos, Nov 18 2006

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

			G.f. = 1 + q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).

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. A002175, A008441, A019669, A053692, A113407, A121444, A122856, A122857, A122865, A125061, A138741, A138745, A138746, A138949, A138950, A138951, A138952, A163746.

Cf. A000122, A000700, A010054, A121373.

s = (EllipticTheta[3, 0, q]^2 + 3*EllipticTheta[3, 0, q^3]^2)/4 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 07 2015, from 2nd formula *)

{a(n) = if( n<1, n==0, sumdiv(n, d, ((d%2) * ((d%3==0)+1)) * (-1)^(d\6)))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
     [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

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

Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) in powers of q.
Expansion of (theta_3(q)^2 + 3*theta_3(q^3)^2) / 4 in powers of q.
Euler transform of period 12 sequence [ 1, 0, 2, -2, 1, -2, 1, -2, 2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1-(-1)^e)/2 if p == 3 (mod 4).
G.f.: 1 + Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122857.
a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 3 * A008441(n). a(6*n + 1) = A002175(n). a(6*n + 5) = 2 * A121444(n). a(8*n + 1) = A113407(n). a(8*n + 3) = 3 * A113407(n). a(8*n + 5) = 2 * A053692(n). a(8*n + 7) = 6 * A053692(n). a(9*n) = A125061(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 24 2023

A138741 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q (unsigned).

Original entry on oeis.org

1, 3, 2, 0, 1, 0, 2, 6, 2, 0, 0, 0, 3, 3, 2, 0, 0, 0, 2, 6, 2, 0, 2, 0, 1, 6, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 9, 0, 0, 1, 0, 4, 6, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 6, 2, 0, 2, 0, 1, 6, 4, 0, 0, 0, 0, 6, 2, 0, 0, 0, 4, 3, 2, 0, 2, 0, 2, 6, 0, 0, 0, 0, 3, 0, 2

Offset: 0

Michael Somos, Mar 27 2008

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

			G.f. = 1 + 3*x + 2*x^2 + x^4 + 2*x^6 + 6*x^7 + 2*x^8 + 3*x^12 + 3*x^13 + ...
G.f. = q + 3*q^3 + 2*q^5 + q^9 + 2*q^13 + 6*q^15 + 2*q^17 + 3*q^25 + 3*q^27 + ...

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. A002175, A008441, A019669, A116604, A121444, A132003.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x]^2 + 3 EllipticTheta[ 2, 0, x^3]^2) / 4, {x, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, # == 3, 2 - (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger[2 n + 1])]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = QP[q^2]^7*QP[q^3]*QP[q^12]^2 / (QP[q]^3*QP[q^4]^2* QP[q^6]^3) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, (-1)^(d\6) * [0, 1, 0, 2, 0, 1][d%6 + 1]))};

{a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2 )))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^3), n))};

Expansion of q^(-1/2) * (theta_2(q)^2 + 3 * theta_2(q^3)^2) / 4 in powers of q.
Expansion of phi(q) * psi(q) * psi(q^3) / phi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -2, 3, -2, 2, -4, 3, -2, ...].
Moebius transform is period 24 sequence [ 1, -1, 2, 0, 1, -2, -1, 0, -2, -1, -1, 0, 1, 1, 2, 0, 1, 2, -1, 0, -2, 1, -1, 0, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1 + (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1+(-1)^e)/2 if p = 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132003.
a(6*n + 3) = a(6*n + 5) = 0.
a(n) = (-1)^n * A116604(n). a(2*n) = A008441(n).
a(6*n) = A002175(n). a(6*n + 1) = 3 * A008441(n). a(6*n + 2) = 2 * A121444(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Dec 28 2023

A053300 Continued fraction for Pi/2.

Original entry on oeis.org

1, 1, 1, 3, 31, 1, 145, 1, 4, 2, 8, 1, 6, 1, 2, 3, 1, 4, 1, 5, 1, 41, 1, 2, 3, 7, 1, 1, 1, 27, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 49, 2, 1, 4, 3, 6, 2, 1, 3, 3, 17, 1, 3, 2, 1, 6, 3, 1, 6, 26, 3, 1, 1, 3, 4, 3, 2, 14, 11, 1, 4, 1, 1, 5, 2, 8, 8, 2, 80, 1, 1, 22, 2, 11, 2, 1

Offset: 0

N. J. A. Sloane, Mar 21 2000

			1.57079632679489661923132169... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(31 + ...)))). - _Harry J. Smith_, May 31 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
Michael A. Filaseta, Allen Stenger, D. Callan and Z. Franco, Solution to Problem 10640: When a Multiple of Pi/2 is Close to an Integer, Amer. Math. Monthly, 107 (2000), 177-178.
I. Rosenholtz, Tangent sequences, world records, ..., Math. Mag., 72 (No. 5, 1999), 367-376.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A001203.

Cf. A019669 (decimal expansion). - Harry J. Smith, May 31 2009

R:= RealField(); ContinuedFraction(Pi(R)/2); // G. C. Greubel, May 24 2018

Mathematica
```
ContinuedFraction[ Pi/2, 100 ]
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi/2); for (n=0, 20000, write("b053300.txt", n, " ", x[n+1])); } \\ Harry J. Smith, May 31 2009

A228824 Decimal expansion of 4*Pi/5.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 03 2013

With offset 2, decimal expansion of 8*Pi.

8*Pi is also the surface area of a sphere whose diameter equals the square root of 8, hence its radius equals the square root of 2. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013

			4*Pi/5 = 2.5132741228718345907701147066236023073577...
8*Pi = 25.132741228718345907701147066236023073577...

Index entries for transcendental numbers

Cf. A002193, A228819.

Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721 (Pi through 7*Pi).

RealDigits[(4 Pi)/5,10,120][[1]] (* Harvey P. Dale, Aug 24 2017 *)

 8*Pi \\ Charles R Greathouse IV, Oct 01 2022

A352324 Decimal expansion of 4Pi / (5sqrt(10-2*sqrt(5))).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 12 2022

Cauchy's residue theorem implies that Integral_{x=0..oo} 1/(1 + x^m) dx = (Pi/m) * csc(Pi/m); this is the case m = 5.

The area of a circle circumscribing a unit-area regular decagon.

			1.0689593321155951134251843725068826399014509252665...

Jean-François Pabion, Éléments d'Analyse Complexe, licence de Mathématiques, page 111, Ellipses, 1995.

Index entries for transcendental numbers.

Cf. A019694, A047209, A121570, A371604, A377405.

Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), this sequence (m=5), A019670 (m=6), A352125 (m=8), A094888 (m=10).

evalf(4*Pi / (5*(sqrt(10-2sqrt(5)))), 100);

First[RealDigits[N[4Pi/(5Sqrt[10-2Sqrt[5]]), 93]]] (* Stefano Spezia, Mar 12 2022 *)

Equals Integral_{x=0..oo} 1/(1 + x^5) dx.
Equals (Pi/5) *csc(Pi/5).
Equals (1/2) * A019694 * A121570.
Equals 1/Product_{k>=1} (1 - 1/(5*k)^2). - Amiram Eldar, Mar 12 2022
Equals Product_{k>=2} (1 + (-1)^k/A047209(k)). - Amiram Eldar, Nov 22 2024
Equals 1/A371604 = A377405/5. - Hugo Pfoertner, Nov 22 2024

A004531 Number of integer solutions to x^2 + 4 * y^2 = n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

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

			G.f. = 1 + 2*x + 4*x^4 + 4*x^5 + 4*x^8 + 2*x^9 + 4*x^13 + 4*x^16 + 4*x^17 + 8*x^20 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 32.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004018, A004020, A019669.

CoefficientList[EllipticTheta[3, 0, x]*EllipticTheta[3, 0, x^4] + O[x]^105, x] (* Jean-François Alcover, Nov 05 2015 *)

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n])}; /* Michael Somos, Jul 04 2005 */

{a(n) = local(A, e1, e2, e4); if( n<0, 0, A = x * O(x^n); e1 = eta(x^2 + A); e2 = eta(x^4 + A); e4 = eta(x^8 + A); polcoeff( (e2^12 + e1^8 * e4^4 + 4 * x * e1^4 * e4^8) / (2 * e1^4 * e2^2 * e4^4), n))};

Q = DiagonalQuadraticForm(ZZ, [4, 1])
Q.representation_number_list(105) # Peter Luschny, Jun 20 2014

Expansion of (eta(q^2) * eta(q^8))^5 / (eta(q)^2 * eta(q^4)^4 * eta(q^16)^2) in powers of q.
Expansion of phi(x) * phi(x^4) = phi(x^4)^2 + 2 * x * psi(x^4)^2 in powers of x where phi(x), psi(x) are Ramanujan theta functions.
Expansion of (theta2^2(q^2) + theta3^2(q^2) + theta4^2(q^2)) / 2 in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, 1, 2, -3, 2, -4, 2, -3, 2, 1, 2, -3, 2, -2, ...]. - Michael Somos, Jun 20 2014
G.f.: Sum_{i,j} x^(i^2 + 4 * j^2).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A004018(n). a(4*n + 1) = A004020(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Oct 15 2022

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