A004018 -id:A004018 - cp's OEIS frontend

A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 2, 0, 12, 8, 1, 0, 0, 4, 8, 24, 10, 1, 0, 0, 8, 6, 32, 40, 12, 1, 0, 0, 0, 24, 24, 80, 60, 14, 1, 0, 0, 0, 24, 48, 90, 160, 84, 16, 1, 0, 2, 4, 0, 96, 112, 252, 280, 112, 18, 1, 0, 0, 4, 12, 64, 240, 312, 574, 448, 144, 20, 1

Offset: 0

Peter Luschny, Oct 01 2018

Number of ways of writing k as a sum of n squares.

			[ 0] 1,  0,    0,    0,     0,     0,     0      0,     0,     0, ... A000007
[ 1] 1,  2,    0,    0,     2,     0,     0,     0,     0,     2, ... A000122
[ 2] 1,  4,    4,    0,     4,     8,     0,     0,     4,     4, ... A004018
[ 3] 1,  6,   12,    8,     6,    24,    24,     0,    12,    30, ... A005875
[ 4] 1,  8,   24,   32,    24,    48,    96,    64,    24,   104, ... A000118
[ 5] 1, 10,   40,   80,    90,   112,   240,   320,   200,   250, ... A000132
[ 6] 1, 12,   60,  160,   252,   312,   544,   960,  1020,   876, ... A000141
[ 7] 1, 14,   84,  280,   574,   840,  1288,  2368,  3444,  3542, ... A008451
[ 8] 1, 16,  112,  448,  1136,  2016,  3136,  5504,  9328, 12112, ... A000143
[ 9] 1, 18,  144,  672,  2034,  4320,  7392, 12672, 22608, 34802, ... A008452
[10] 1, 20,  180,  960,  3380,  8424, 16320, 28800, 52020, 88660, ... A000144
   A005843,   v, A130809,  v,  A319576,  v ,   ...      diagonal: A066535
           A046092,    A319575,       A319577,     ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954.
J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006).

Seiichi Manyama, Descending antidiagonals n = 0..139, flattened
L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124.
S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Index entries for sequences related to sums of squares

Variant starting with row 1 is A122141, transpose of A286815.

A319574row := proc(n, len) series(JacobiTheta3(0, x)^n, x, len+1);
[seq(coeff(%, x, j), j=0..len-1)] end:
seq(print([n], A319574row(n, 10)), n=0..10);
# Alternative, uses function PMatrix from A357368.
PMatrix(10, n -> A000122(n-1)); # Peter Luschny, Oct 19 2022

A[n_, k_] := If[n == k == 0, 1, SquaresR[n, k]];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

for n in (0..10):
    Q = DiagonalQuadraticForm(ZZ, [1]*n)
    print(Q.theta_series(10).list())

A112610 Number of representations of n as a sum of two squares and two triangular numbers.

Original entry on oeis.org

1, 6, 13, 14, 18, 32, 31, 30, 48, 38, 42, 78, 57, 54, 80, 62, 84, 96, 74, 96, 121, 108, 90, 128, 98, 102, 192, 110, 114, 182, 133, 156, 176, 160, 138, 192, 180, 150, 234, 158, 192, 288, 183, 174, 240, 182, 228, 320, 194, 198, 272, 252, 240, 288, 256, 252, 403, 230

Offset: 0

James Sellers, Dec 21 2005

nonn

Also row sums of A239931, hence the sequence has a symmetric representation. - Omar E. Pol, Aug 30 2015

			a(1) = 6 since we can write 1 = 1^2 + 0^2 + 0 + 0 = (-1)^2 + 0^2 + 0 + 0 = 0^2 + 1^2 + 0 + 0 = 0^2 + (-1)^2 + 0 + 0 = 0^2 + 0^2 + 1 + 0 = 0^2 + 0^2 + 0 + 1

Amiram Eldar, Table of n, a(n) for n = 0..10001
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

Cf. A000203, A193553, A237593, A239052, A239053, A239931.

[DivisorSigma(1, 4*n+1): n in [0..60]]; // Vincenzo Librandi, Sep 18 2015

Table[DivisorSigma[1, 4 n + 1], {n, 0, 57}] (* Michael De Vlieger, Aug 31 2015 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^14/eta(x+A)^6/eta(x^4+A)^4, n))} /* Michael Somos, Jul 04 2006 */

a(n) = sigma(4n+1) where sigma(n) = A000203(n) is the sum of the divisors of n.
Euler transform of period 4 sequence [ 6, -8, 6, -4, ...]. - Michael Somos, Jul 04 2006
Expansion of q^(-1/4)eta^14(q^2)/(eta^6(q)eta^4(q^4)) in powers of q. - Michael Somos, Jul 04 2006
Expansion of psi(q)^2*phi(q)^2, i.e., convolution of A004018 and A008441 [Hirschhorn]. - R. J. Mathar, Mar 24 2011
Sum_{k=0..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A305575 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

Original entry on oeis.org

0, 1, 0, -1, 0, 1, -1, -1, 1, 2, 0, -2, 0, 2, 1, -1, -2, -2, -1, 1, 2, 2, -2, -2, 2, 3, 0, -3, 0, 3, 1, -1, -3, -3, -1, 1, 3, 3, 2, -2, -3, -3, -2, 2, 3, 4, 0, -4, 0, 4, 1, -1, -4, -4, -1, 1, 4, 3, -3, -3, 3, 4, 2, -2, -4, -4, -2, 2, 4, 5, 4, 3, 0, -3, -4, -5, -4, -3, 0, 3, 4, 5, 1, -1

Offset: 0

Hugo Pfoertner, Jun 05 2018

Similar to A283307, but with secondary sorting by polar angle.

			The first points (listing [polar angle phi,x,y]) are:
r^2
  0: [0.0*Pi,0,0];
  1: [0.0*Pi,1,0], [0.5*Pi,0,1], [1.0*Pi,-1,0], [1.5*Pi,0,-1];
  2: [0.25*Pi,1,1], [0.75*Pi,-1,1], [1.25*Pi,-1,-1], [1.75*Pi,1,-1];
  4: [0.0*Pi,2,0], [0.5*Pi,0,2], [1.0*Pi,-2,0], [1.5*Pi,0,-2];
  5: [0.148*Pi,2,1], [0.352*Pi,1,2], [0.648*Pi,-1,2], [0.852*Pi,-2,1],
   [1.148*Pi,-2,-1], [1.352*Pi,-1,-2], [1.648*Pi,1,-2], [1.852*Pi,2,-1];
  8: [0.25*Pi,2,2], [0.75*Pi,-2,2], [1.25*Pi,-2,-2], [1.75*Pi,2,-2].

Hugo Pfoertner, Table of n, a(n) for n = 0..17664 (covering range r <= 75).
Hugo Pfoertner, Illustration of A305576 vs A305575.
Rainer Rosenthal, Coloured Circular Visualization of Spiral [A305575,A305576]

For the y-coordinates see A305576.

Cf. A000328, A004018, A283307, A283308, A307014, A307016, A307017, A308080, A308081.

atan2(y,x)=if(x>0,atan(y/x),if(x==0,if(y>0,Pi/2,-Pi/2),if(y>=0,atan(y/x)+Pi,atan(y/x)-Pi)));
angle(x,y)=(atan2(y,x)+2*Pi)%(2*Pi);
{a004018(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))};
xyselect=1; \\ change to 2 for A305576
print1(0,", ");for(s=1,25,my(r=a004018(s));if(r>0,my(v=matrix(r,3),w=vector(r),m=sqrtint(s),L=0);for(i=-m,m,my(k=s-i^2,kk);if(k==0,v[L++,1]=i;v[L,2]=0;v[L,3]=angle(i,0),if(issquare(k),kk=sqrtint(k);forstep(j=-kk,kk,kk+kk,v[L++,1]=i;v[L,2]=j;v[L,3]=angle(i,j)))));p=vecsort(v[,3],,1);for(k=1,L,w[k]=v[p[k],xyselect]);for(k=1,L,print1(w[k],", ")))); \\ Hugo Pfoertner, May 12 2019

A305576 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives y-coordinates.

Original entry on oeis.org

0, 0, 1, 0, -1, 1, 1, -1, -1, 0, 2, 0, -2, 1, 2, 2, 1, -1, -2, -2, -1, 2, 2, -2, -2, 0, 3, 0, -3, 1, 3, 3, 1, -1, -3, -3, -1, 2, 3, 3, 2, -2, -3, -3, -2, 0, 4, 0, -4, 1, 4, 4, 1, -1, -4, -4, -1, 3, 3, -3, -3, 2, 4, 4, 2, -2, -4, -4, -2, 0, 3, 4, 5, 4, 3, 0, -3, -4, -5, -4, -3, 1

Offset: 0

Hugo Pfoertner, Jun 05 2018

Similar to A283308, but with secondary sorting by polar angle.

			See A305575.

Rainer Rosenthal, Table of n, a(n) for n = 0..17664, covering range r <= 75 (terms 0..7844 from Hugo Pfoertner).

For the x-coordinates see A305575.

Cf. A000328, A004018, A283307, A283308, A307014, A307016, A307017, A308080, A308081.

PARI
```
\\ Provided in A305575.
```

A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

Original entry on oeis.org

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 27625, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125

Offset: 1

N. J. A. Sloane

nonn

a(n) is least term of A054994 with exactly n divisors. - Ray Chandler, Jan 05 2012
From Jianing Song, Apr 24 2021: (Start)
a(n) is the smallest k such that A004018(k) = 4n.
Also a(n) is the smallest index of n in A002654.
a(n) is the smallest term in A004613 that has exactly n divisors.
This is a subsequence of A054994. (End)

			4225 = 5^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 4 with exactly 9 divisors, so a(9) = 4225. - _Jianing Song_, Apr 24 2021

Ray Chandler, Table of n, a(n) for n = 1..1432 (a(1433) exceeds 1000 digits).

Cf. A002144, A002654, A004018, A005179, A054994, A000446, A124980, A016032, A093195.

(* This program is not convenient to compute huge terms - A054994 is assumed to be computed with maxTerm = 10^16 *) a[n_] := Catch[ For[k = 1, k <= Length[A054994], k++, If[DivisorSigma[0, A054994[[k]]] == n, Throw[A054994[[k]]]]]]; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jan 21 2013, after Ray Chandler *)

primelist(d,r,l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p,d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A018782(n)=my(pf=prodR(n, n), a=1, b, v=primelist(4, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bJianing Song, Apr 25 2021, following R. J. Mathar's program for A005179.

A000446(n) = min(a(2n-1), a(2n)) for n > 1.
A124980(n) = min(a(2n-1), a(2n)).
A016032(n) = min(2*a(2n-1), a(2n), a(2n+1)).
A093195(n) = min(a(2n), a(2n+1)).
From Jianing Song, Apr 24 2021: (Start)
If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 4. The smallest n such that the equality does not hold is n = 16.
a(n) <= 5^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 5^(p-1) * 13^(q-1) for primes p >= q. (End)

A028609 Expansion of (theta_3(z)theta_3(11z) + theta_2(z)theta_2(11z)).

Original entry on oeis.org

1, 2, 0, 4, 2, 4, 0, 0, 0, 6, 0, 2, 4, 0, 0, 8, 2, 0, 0, 0, 4, 0, 0, 4, 0, 6, 0, 8, 0, 0, 0, 4, 0, 4, 0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 2, 12, 0, 4, 4, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 8, 0, 0, 0, 2, 0, 0, 4, 0, 8, 0, 4, 0, 0, 0, 12, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 8, 0, 0, 0, 4, 0, 6, 6, 0, 0

Offset: 0

N. J. A. Sloane

Theta series of lattice with Gram matrix [2, 1; 1, 6].
Number of integer solutions (x, y) to x^2 + x*y + 3*y^2 = n. - Michael Somos, Sep 20 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^9 + 2*x^11 + 4*x^12 + 8*x^15 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 6] = 1 + 2*q^2 + 4*q^6 + 2*q^8 + 4*q^10 + 6*q^18 + 2*q^22 + 4*q^24 + 8*q^30 + 2*q^32 + 4*q^40 + 4*q^46 + 6*q^50 + 8*q^54 + 4*q^62 + 4*q^66 + 6*q^72 + 4*q^74 + ...

Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

John Cannon, Table of n, a(n) for n = 0..5000
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. A028610, A035179, A028954, A056874.
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), this sequence (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(11), 1), 103); A[1] + 2*A[2] + 4*A[4] + 2*A[5]; /* Michael Somos, Jul 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -11, #] &] 2]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^11], {q, 0, n}]; (* Michael Somos, Jul 12 2014 *)

{a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n * 4\11), 2 * issquare( t=4*n - 11*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep( [ 2, 1; 1, 6], n, 1)), n))}; /* Michael Somos, Apr 21 2015 */

{a(n) = if( n<1, n==0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -11, p) * X))[n] * 2)}; /* Michael Somos, Jun 05 2005 */

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -11, d)))}; /* Michael Somos, Jan 29 2007 */

Expansion of phi(x) * phi(x^11) = 4 * x^3 * psi(x^2) * psi(x^22) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 21 2015
From Michael Somos, Jan 29 2007: (Start)
Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...].
a(n) = 2 * b(n) and b(n) is multiplicative with b(11^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), b(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11).
G.f.: 1 + 2 * Sum_{k>0} Kronecker( -11, k) * x^k / (1 - x^k). (End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 05 2007
Expansion of (F(x)^2 + 4 * F(x^2)^2 + 8 * F(x^4)^2) / F(x^2) in powers of x or expansion of (F(x)^2 + 2 * F(x^2)^2 + 2 * F(x^4)^2) / F(x^2) in powers of x^4 where F(x) = x^(1/2) * f(-x) * f(-x^11) and f() is a Ramanujan theta function. - Michael Somos, Mar 01 2010
a(n) = 2 * A035179(n) unless n=0. Convolution square is A028610.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(11) = 1.894451... . - Amiram Eldar, Dec 16 2023

A101606 a(n) = number of divisors of n that have exactly three (not necessarily distinct) prime factors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 3, 0, 0, 0, 2, 0, 3, 0, 1, 0, 0, 0, 2, 0, 1, 1, 2, 0, 1, 0, 2, 1

Offset: 1

Jonathan Vos Post, Dec 09 2004

This is the inverse Moebius transform of A101605. If n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=3}| + ((j-1)/2)*|{k: ek>=2}| + C(j,3). The first term is the number of distinct cubes of primes in the factors of n (the first way of finding a 3-almost prime). The second term is the number of distinct squares of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 3-almost prime). The third term is the number of distinct products of 3 distinct primes, which is the number of combinations of j primes taken 3 at a time, A000292(j), (the third way of finding a 3-almost prime).

			a(60) = 3 because of all the divisors of 60 only these three are terms of A014612: 12 = 2 * 2 * 3; 20 = 2 * 2 * 5; 30 = 2 * 3 * 5.

Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Antti Karttunen, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Moebius Transform.
Index entries for sequences computed from exponents in factorization of n

Cf. A101605, A014612, A001358, A064911, A001221, A000005, A000010, A004018, A000292.

isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: A101606 := proc(n) a :=0 ; for d in numtheory[divisors](n) do if isA014612(d) then a := a+1 ; fi; od: a ; end: for n from 1 to 120 do printf("%d,",A101606(n)) ; od: # R. J. Mathar, Jan 27 2009

a[n_] := DivisorSum[n, Boole[PrimeOmega[#] == 3]&];
Array[a, 105] (* Jean-François Alcover, Nov 14 2017 *)

A101606(n) = sumdiv(n,d,(3==bigomega(d))); \\ Antti Karttunen, Jul 23 2017

If n = (p1^e1 * p2^e2 * ... * pj^ej) for primes p1, p2, ..., pj and integer exponents e1, e2, ..., ej, then a(n) = a(n) = |{k: ek>=3}| + ((j-1)/2)*|{k: ek>=2}| + C(j, 3). where C(j, 3) is the binomial coefficient A000292(j).

a(n) = Sum_{d|n} A101605(d). - Antti Karttunen, Jul 23 2017

a(48) replaced with 2 and a(76) replaced with 1 by R. J. Mathar, Jan 27 2009

Name changed by Antti Karttunen, Jul 23 2017

A063730 Number of solutions to w^2 + x^2 + y^2 + z^2 = n in positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 6, 0, 4, 4, 0, 12, 1, 0, 12, 4, 6, 4, 12, 12, 0, 12, 6, 12, 12, 0, 24, 16, 0, 12, 18, 12, 13, 16, 12, 28, 6, 0, 36, 16, 12, 24, 24, 24, 4, 16, 30, 24, 18, 12, 36, 36, 0, 28, 42, 12, 36, 16, 24, 52, 1, 24, 48, 28, 18, 24, 60, 36, 12

Offset: 0

N. J. A. Sloane, Aug 23 2001

nonn

Cf. A063691, A063725.

Column k=4 of A337165.

r[n_] := Reduce[ w > 0 && x > 0 && y > 0 && z > 0 && w^2 + x^2 + y^2 + z^2 == n, {w, x, y, z}, Integers]; a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === Or, Length[rn], True, 1]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jul 22 2013 *)
a[n_ ] := Length[FindInstance[{n == w^2 + x^2 + y^2 + z^2, w > 0, x > 0, y > 0, z > 0}, {w, x, y, z}, Integers, 10^18]]; (* Michael Somos, Jun 23 2023 *)

seq(n)=Vec((sum(k=1, sqrtint(n), x^(k^2)) + O(x*x^n))^4 + O(x*x^n), -(n+1)) \\ Andrew Howroyd, Aug 08 2018

G.f.: (Sum_{m>=1} x^(m^2))^4.
a(n) = ( A000118(n) - 4*A005875(n) + 6*A004018(n) - 4*A000122(n) + A000007(n) )/16. - Max Alekseyev, Sep 29 2012
G.f.: ((theta_3(q) - 1)/2)^4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A104794 Expansion of theta_4(q)^2 in powers of q.

Original entry on oeis.org

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8

Offset: 0

Michael Somos, Mar 26 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).
In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A000729, A001934, A002131, A004018, A008441, A053692, A113407, A113652, A122856, A122865, A204531, A227695, A246950, A256014, A258210.

# JacobiTheta4 is defined in A002448.
A104794List(len) = JacobiTheta4(len, 2)
A104794List(102) |> println # Peter Luschny, Mar 12 2018

A := Basis( ModularForms( Gamma1(8), 1), 100); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 4 DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)

{a(n) = if( n<1, n==0, (-1)^n * 4 * sumdiv(n, d, (d%4==1) - (d%4==3)))};

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

{a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))};

Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (1-k^2)^(1/2) K(k^2) / (Pi/2) in powers of q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind.
Expansion of  K(k^2) / (Pi/2) in powers of -q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind. - Michael Somos, Jun 08 2015
Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -4, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.
Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008441.
G.f.: theta_4(q)^2 = (Sum_{k in Z} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.
G.f.: 1 + 4 * Sum_{k>0} (-x)^k / (1 + x^(2*k)). - Michael Somos, Jun 08 2015
a(4*n + 3) = 0. a(n) = (-1)^n * A004018(n) = a(2*n). a(4*n + 1) = -4 * A008441(n). a(n) = -4 * A113652(n) unless n=0. a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = 4 * A122856(n). a(8*n + 1) = -4 * A113407(n). a(8*n + 5) = -8 * A053692(n).
a(n) = a(9*n) = A204531(8*n) = A246950(8*n) = A256014(9*n) = A258210(n). - Michael Somos, Jun 08 2015
Convolution inverse of A001934. Convolution with A000729 is A227695. - Michael Somos, Jun 08 2015
G.f.: 2 * Sum_{k in Z} (-1)^k * x^(k*(k + 1)/2) / (1 + x^k). - Michael Somos, Nov 05 2015
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A057961 Number of points in square lattice covered by a disc centered at (0,0) as its radius increases.

Original entry on oeis.org

1, 5, 9, 13, 21, 25, 29, 37, 45, 49, 57, 61, 69, 81, 89, 97, 101, 109, 113, 121, 129, 137, 145, 149, 161, 169, 177, 185, 193, 197, 213, 221, 225, 233, 241, 249, 253, 261, 277, 285, 293, 301, 305, 317, 325, 333, 341, 349, 357, 365, 373, 377, 385, 401, 405, 421

Offset: 1

Ken Takusagawa, Oct 15 2000

Useful for rasterizing circles.

Conjecture: the number of lattice points in a quadrant of the disk is equal to A000592(n-1). - L. Edson Jeffery, Feb 10 2014

			a(2)=5 because (0,0); (0,1); (0,-1); (1,0); (-1,0) are covered by any disc of radius between 1 and sqrt(2).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

T. D. Noe, Table of n, a(n) for n=1..1000
L. Edson Jeffery, Illustration of first few terms.

Cf. A004018, A004020, A005883, A057962. Distinct terms of A057655.

Cf. A000404, A001481, A232499.

max = 100; A001481 = Select[Range[0, 4*max], SquaresR[2, #] != 0 &]; Table[SquaresR[2, A001481[[n]]], {n, 1, max}] // Accumulate (* Jean-François Alcover, Oct 04 2013 *)

cp's OEIS Frontend

A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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A112610 Number of representations of n as a sum of two squares and two triangular numbers.

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A305575 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

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A305576 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives y-coordinates.

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A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

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A028609 Expansion of (theta_3(z)*theta_3(11z) + theta_2(z)*theta_2(11z)).

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A101606 a(n) = number of divisors of n that have exactly three (not necessarily distinct) prime factors.

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A028609 Expansion of (theta_3(z)theta_3(11z) + theta_2(z)theta_2(11z)).