A004018 -id:A004018 - cp's OEIS frontend

A062089 Decimal expansion of Sierpiński's constant.

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

Offset: 1

Jason Earls, Jun 27 2001

			2.5849817595792532170658935873831711600880516518526309173215...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 122-126.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Steven R. Finch, Sierpinski's Constant. [Broken link]
Steven R. Finch, Sierpinski's Constant. [From the Wayback machine]
Simon Plouffe, Sierpinski Constant to 2000 digits. [Note: Last 2 digits given at this link are incorrect. - _William Echols_, Jun 04 2025]
Wacław Sierpiński, O sumowaniu szeregu Sigma_{n>a}^{n<=b} tau(n) f(n), gdzie tau(n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych, Prace Matematyczno-Fizyczne, Vol. 18, No. 1 (1907), pp. 1-59.
Eric Weisstein's World of Mathematics, Sierpiński Constant.
Wikipedia, Sierpiński's constant.

Cf. A004018, A062083, A222882, A222883.

K=-Pi Log[Pi]+2 Pi EulerGamma+4 Pi Log[Gamma[3/4]];First@RealDigits[N[K,105]](* Ant King, Mar 02 2013 *)

-Pi*log(Pi)+2*Pi*Euler+4*Pi*log(gamma(3/4))

{ default(realprecision, 5080); x=-Pi*log(Pi)+2*Pi*Euler+4*Pi*log(gamma(3/4)); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062089.txt", n, " ", d)) } \\ Harry J. Smith, Aug 01 2009

Equals -Pi*log(Pi)+2*Pi*gamma+4*Pi*log(GAMMA(3/4)).
Equals Pi*A241017. - Eric W. Weisstein, Dec 10 2014
Equals Pi*(A086058-1). - Eric W. Weisstein, Dec 10 2014
Equals lim_{n->oo} (A004018(n)/n - Pi*log(n)). - Amiram Eldar, Apr 15 2021

A068785 Number of Cartesian lattice points in or on the circle x^2 + y^2 = 10^n.

Original entry on oeis.org

5, 37, 317, 3149, 31417, 314197, 3141549, 31416025, 314159053, 3141592409, 31415925457, 314159264013, 3141592649625, 31415926532017, 314159265350589, 3141592653588533, 31415926535867961, 314159265358987341, 3141592653589764829, 31415926535897744669

Offset: 0

Robert G. Wilson v, Mar 07 2002

nonn

a(n) ~ Pi*10^n [Shanks, page 164]. "Gauss gave [a(2)] = 317 and [a(4)] = 31417." [Shanks, page 165].

Daniel Shanks, "Solved and Unsolved Problems in Number Theory," Fourth Edition, Chelsea Publishing Co., NY, 1993, pages 164-165 and 234 [gives a(n) for n = 8, 10, 12, 14].
Wolfram Research, Mathematica 4, Standard Add-On Packages, Wolfram Media, Inc., Champaign, Il, 1999, pages 322-3.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36
Index entries for sequences related to populations of quadratic forms

Cf. A004018, A057961, A057655.

k = 1; s = 1; Do[s = s + SquaresR[2, n]; If[n == 10^k, k++; Print[s]], {n, 1, 10^6} ]

a(n) = Sum_{k=0..10^n} A004018(k). - Robert Israel, Jul 13 2014

Definition and comments corrected by Jonathan Sondow, Dec 28 2012

a(0) corrected and a(9)-a(19) from Hiroaki Yamanouchi, Jul 13 2014

A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 864, 1024, 1152, 1280, 1536, 1728, 2048, 2304, 2560, 3072, 3456, 3840, 4096, 4608, 5120, 6144, 6912, 7680, 8192, 9216, 10240, 11520, 12288, 13824, 15360

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

			Circles with radius 1 and 2 have 4 lattice points on their circumference, so a(1)=4. A circle with radius sqrt(5) passes through 8 lattice points of the shape (2,1), so a(2)=8. A circle with radius 5 passes through 4 lattice points of shape (5,0) and through 8 points of shape (4,3), so a(3)=4+8=12
A071383(11) = 5^2 * 13^2 * 17^1 = 71825. Therefore A071385(11) = 4*(2+1)*(2+1)*(1+1) = 72.

Ray Chandler, Table of n, a(n) for n = 1..425
Hugo Pfoertner, Construction of A071383, A071384, A071385

Records in A004018. Cf. A071383, A071384, A002144, A002654, A300163, A344470.

my(v=list(10^15), rec=0); print1(1, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(4*rec, ", "))) \\ Jianing Song, May 20 2021, see program for A054994

a(n) = 4 * Product_{k=1..klim} (e_k + 1), where klim and e_1 >= e_2 >= ... >= e_klim > 0 are known from A071383(n) = Product_{k=1..klim} p_k^e_k, with p_k = k-th prime of the form 4i+1. (J. H. Conway)
a(n) = 4*A000005(A071383(n)) for n > 1.
a(n) = A004018(A071383(n)).
a(n) = A002654(A071383(n)) for n > 1. - Jianing Song, May 20 2021
a(n) = 4*A344470(n-1) for n > 1. - Hugo Pfoertner, Sep 04 2022

A133675 Negative discriminants with form class number 1 (negated).

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163

Offset: 1

N. J. A. Sloane, May 16 2003

The list on p. 260 of Cox is missing -12, the list in Theorem 7.30 on p. 149 is correct. - Andrew V. Sutherland, Sep 02 2012

Let b(k) be the number of integer solutions of f(x,y) = k, where f(x,y) is the principal binary quadratic form with discriminant d<0 (i.e., f(x,y) = x^2 - (d/4)*y^2 if 4|d, x^2 + x*y + ((1-d)/4)*y^2 otherwise), then this sequence lists |d| such that {b(k)/b(1): k>=1} is multiplicative. See Crossrefs for the actual sequences. - Jianing Song, Nov 20 2019

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, pp. 149, 260.
D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Cf. A014602, A003173.
The sequences {b(k): k>=0}: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A033716 (d=-12), A004531 (d=-16), A028641 (d=-19), A138805 (d=-27), A033719 (d=-28), A138811 (d=-43), A318984 (d=-67), A318985 (d=-163).
The sequences {b(k)/b(1): k>=1}: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A096936 (d=-12), A113406 (d=-16), A035171 (d=-19), A138806 (d=-27), A110399 (d=-28), A035147 (d=-43), A318982 (d=-67), A318983 (d=-163).

ok(n)={(-n)%4<2 && quadclassunit(-n).no == 1} \\ Andrew Howroyd, Jul 20 2018

Corrected by David Brink, Dec 29 2007

A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

The number of integer solutions (x, y) to x^2 + x*y + 5*y^2 = n, discriminant -19. - Ray Chandler, Jul 12 2014

			G.f. = 1 + 2*x + 2*x^4 + 4*x^5 + 4*x^7 + 2*x^9 + 4*x^11 + 2*x^16 + 4*x^17 + 2*x^19 + ...

Robert Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 409, Eq. (19).

G. C. Greubel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035171.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), this sequence (d=-19), A138811 (d=-43).

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -19, #] &] 2]; (* Michael Somos, Jun 14 2012 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-19, d)) * 2)}; /* Michael Somos, Feb 27 2007 */

{a(n) = if( n<1, n==0, qfrep([2, 1;1, 10], n, 1)[n] * 2)}; /* Michael Somos, Feb 27 2007 */

Theta series of quadratic form with Gram matrix [ 2, 1; 1, 10 ].
Expansion of phi(q) * phi(q^19) + 4 * q^5 * psi(q^2)* psi(q^38) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 27 2007
Moebius transform is period 19 sequence [2, -2, -2, 2, 2, 2, 2, -2, 2, -2, 2, -2, -2, -2, -2, 2, 2, -2, 0, ...]. - Michael Somos, Feb 27 2007
a(n) = 2*b(n) where b(n) is multiplicative with a(0) = 1,  b(19^e) = 1, b(p^e) = e + 1 if Kronecker(-19, p) = 1, b(p^e) = (1 + (-1)^e)/2 if Kronecker(-19, p) = -1. - Michael Somos, Feb 27 2007
a(n) = 2 * A035171(n) unless n = 0. - Jianing Song, Sep 06 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(19) = 1.441461... . - Amiram Eldar, Dec 16 2023

A138811 Theta series of quadratic form x^2 + xy + 11y^2.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 4, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 4, 0, 0, 0

Offset: 0

Michael Somos, Mar 31 2008, Apr 05 2008

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

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035147.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), this sequence (d=-43).

A := Basis( ModularForms( Gamma1(43), 1), 87); A[1] + 2*A[2] + 2*A[5] + 2*A[10] + 4*A[12] + 4*A[14] + 2*A[17] + 4*A[18]; /* Michael Somos, Sep 07 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -43, #] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^43] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^43], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-43, #]&]];
2 Table[a[n], {n, 1, 100}]] (* Vincenzo Librandi, Sep 07 2018 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))};

a(n)=if(n, sumdivmult(n,d,kronecker(-43,d))*2, 0) \\ Charles R Greathouse IV, Nov 23 2021

Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
Expansion of phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 43 sequence [2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(43^e) = 1, b(p^e) = e + 1 if Kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
G.f.: Sum_{i,j in Z} x^(i*i + i*j + 11*j*j).
a(n) = 2 * A035147(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(43) = 0.958176... . - Amiram Eldar, Nov 21 2023

A071339 a(n) = the maximum number of lattice points touched by an origin-centered circle with radius <= n.

Original entry on oeis.org

1, 4, 4, 8, 8, 12, 12, 12, 12, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 36, 36, 36

Offset: 0

Hugo Pfoertner, May 22 2002

			a(5 to 8)=12 because the circle with radius 5 touches 12 lattice points: (+-1,+-2), (+-2,+-1), (+-5,0), (0,+-5); but no circle touches more, until the radius is sqrt(65).

Hugo Pfoertner, FORTRAN program for A071339-A071347

Cf. A004018, A071339-A071347, A071383-A071385.

a(n) = max(i=0 to n^2) A004018(i)

Edited by Don Reble, Nov 06 2005

A258228 Expansion of f(q) * f(-q^2) * chi(q^3) in powers of q where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -2, 0, 1, -4, 0, 0, -2, 4, 2, 0, 0, 2, 0, 0, 1, -4, 4, 0, -4, 0, 0, 0, 0, 3, -4, 0, 0, -4, 0, 0, -2, 0, 2, 0, 4, 2, 0, 0, 2, -4, 0, 0, 0, 8, 0, 0, 0, 1, -6, 0, 2, -4, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, -8, 0, 0, -4, 0, 0, 0, 4, 2, -4, 0, 0, 0, 0, 0, -4, 4

Offset: 0

Michael Somos, May 23 2015

sign

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

			G.f. = 1 + q - 2*q^2 + q^4 - 4*q^5 - 2*q^8 + 4*q^9 + 2*q^10 + 2*q^13 + ...

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

Cf. A002175, A004018, A122856, A122865, A258210, A258279, A258292.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] + A[2] - 2*A[3] + A[5] - 4*A[6] - 2*A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] - 4*A[18] + 4*A[19];

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^2 / (QPochhammer[ -q, q^6] QPochhammer[ -q^5, q^6]), {q, 0, n}];

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

Expansion of f(q)^2 * f(-q^6) / f(q, q^5) in powers of q where f(,) is Ramanujan's general theta function.
Expansion of eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -4, 1, -2, 2, -3, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122865.
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 * (1 + x^(3*k)) / ((1 + x^(2*k)) * (1 + x^(6*k))).
a(n) = (-1)^n * A258210(n) = A258279(2*n) = A258292(2*n).
a(3*n + 1) = A122865(n). a(3*n + 2) = -2 * A122856(n). a(9*n) = A004018(n). a(9*n + 3) = a(9*n + 6) = 0.
a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(12*n + 1) = A002175(n).

A345436 Represent the ring of Gaussian integers E = {x+y*i: x, y rational integers, i = sqrt(-1)} by the cells of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells representing the ring identities 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 2, 4, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 59, 61, 67, 69, 75, 77, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The cell with spiral index m represents the Gaussian integer A174344(m+1) + A274923(m+1) * i. So the set of Gaussian primes is {A174344(a(n)+1) + A274923(a(n)+1) * i : n >= 2}. - Peter Munn, Aug 02 2021
The Gaussian integer z = x+i*y has norm x^2+y^2. There are four units (of norm 1), +-1, +-i. The number of Gaussian integers of norm n is A004018(n).
The norms of the Gaussian primes are listed in A055025, and the number of primes with a given norm is given in A055026.
The successive norms of the Gaussian integers along the square spiral are listed in A336336.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.2, p. 106.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

Eric Weisstein's World of Mathematics, Gaussian prime.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Fig. 2.

Equals A308412 - 1. Cf. A345435, A345437.

Cf. also A003173, A004018, A055025, A055026, A103431, A174344, A274923, A336336.

Name clarified by Peter Munn, Aug 02 2021

A194094 Expansion of (2/Pi)*elliptic_E(k) in powers of q.

Original entry on oeis.org

1, -4, 20, -64, 164, -392, 896, -1920, 3908, -7684, 14632, -27072, 48896, -86408, 149760, -255104, 427652, -706568, 1152020, -1855296, 2954056, -4654080, 7260288, -11221632, 17194496, -26131980, 39409960, -59003008, 87728640, -129586568, 190226176, -277587456, 402779396, -581276160, 834539560, -1192216320

Offset: 0

Joerg Arndt, Aug 15 2011

sign

Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = prod(n>=1, 1-q^(k*n) ) (s=k^2 where k is elliptic k), then the g.f. is hypergeom([-1/2, +1/2], [+1], s) (expansion of 2/Pi*elliptic_E(k) in powers of q).

			E(k(q)) = 1 - 4*q + 20*q^2 - 64*q^3 + 164*q^4 - 392*q^5 + 896*q^6 - 1920*q^7 +- ...

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A004018 (elliptic K(k(q))), A115977 (elliptic k(q)^2), A193219.

N:= 100: # to get a(0) to a(N)
t3:= curry(JacobiTheta3,0):
t4:= curry(JacobiTheta4,0):
Dq:= f -> q*diff(f,q):
E1:= t3(q)^2:
E2a:= - 2*(t4(q)/t3(q))^2:
E2b:= t4(q)^(-2):
S1:= series(E1,q,N+1):
S2a:= series(E2a,q,N+1):
S2b:= series(Dq(series(E2b,q,N+1)),q,N+1):
S:= series(S1+S2a*S2b,q,N+1):
seq(coeff(S,q,j),j=0..N); # Robert Israel, Sep 02 2015

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticE[m] / (Pi/2), {q, 0, n}]] (* Michael Somos, Jan 24 2012 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ -1/2, 1/2, 1, ModularLambda[ Log[q] / (Pi I)]], {q, 0, n}] (* Michael Somos, Jan 24 2012 *)
nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[(EllipticTheta[4, 0, x]^4 * EllipticTheta[3, 0, x] + 4*x*dtheta) / EllipticTheta[3, 0, x]^3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)
nmax = 30; CoefficientList[4 q D[Log[EllipticTheta[2,0,q]+O[q]^nmax],q]/(EllipticTheta[3,0,q]+O[q]^nmax)^2,q] (* Mamuka Jibladze, Jun 02 2025 *)

Expansion of theta_3(q)^2 - 2 * (theta_4(q) / theta_3(q))^2 * Dq ( theta_4(q)^-2 ) = theta_3(q)^2 + 4 Dq (theta_4(q)) / (theta_4(q) * theta_3(q)^2) in powers of q where Dq (f) := q * df/dq. - Michael Somos, Jan 24 2012
Expansion of (T4^4 * T3 + 4*q * d/dq T3) / T3^3 where T3 = theta_3(q) and T4 = theta_4(q). - Joerg Arndt, Sep 02 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (Pi * 2^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2019
Expansion of 4 * Dq(theta_2(q))/(theta_2(q) * theta_3(q)^2) in powers of q where Dq(f):= q*df/dq. - Mamuka Jibladze, Jun 02 2025

cp's OEIS Frontend

A062089 Decimal expansion of Sierpiński's constant.

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A068785 Number of Cartesian lattice points in or on the circle x^2 + y^2 = 10^n.

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A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

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A133675 Negative discriminants with form class number 1 (negated).

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A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.

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A138811 Theta series of quadratic form x^2 + x*y + 11*y^2.

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A071339 a(n) = the maximum number of lattice points touched by an origin-centered circle with radius <= n.

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A138811 Theta series of quadratic form x^2 + xy + 11y^2.