A004018 -id:A004018 - cp's OEIS frontend

A063725 Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n.

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

Offset: 0

N. J. A. Sloane, Aug 23 2001

nonn

a(A018825(n))=0; a(A000404(n))>0; a(A081324(n))=1; a(A004431(n))>1. - Reinhard Zumkeller, Aug 16 2011

			a(5) = 2 from the solutions (1,2) and (2,1).

Cf. A000404 (the numbers n that can be represented in this form).
Cf. A000161, A063691, A063730, A025426, A000290, A010052.
Column k=2 of A337165.

a063725 n =
   sum $ map (a010052 . (n -)) $ takeWhile (< n) $ tail a000290_list
a063725_list = map a063725 [0..]
-- Reinhard Zumkeller, Aug 16 2011

nn = 100; t = Table[0, {nn}]; s = Sqrt[nn]; Do[n = x^2 + y^2; If[n <= nn, t[[n]]++], {x, s}, {y, s}]; Join[{0}, t] (* T. D. Noe, Apr 03 2011 *)

a(n)=if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ Charles R Greathouse IV, May 18 2016

from math import prod
from sympy import factorint
def A063725(n):
    f = factorint(n)
    return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items())-(not any(e&1 for e in f.values())) if n else 0 # Chai Wah Wu, May 17 2023

G.f.: (Sum_{m=1..inf} x^(m^2))^2.
a(n) = ( A004018(n) - 2*A000122(n) + A000007(n) )/4. - 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

A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

Original entry on oeis.org

1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36

Offset: 0

Eric W. Weisstein

Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012

The indices of terms not equaling 4 or 12 correspond to A009177, n>0. - Bill McEachen, Aug 14 2025

			a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452.

Cf. A000328, A051132, A009177.

a046109 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
-- Reinhard Zumkeller, Jan 23 2012

N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
  A[x]:= A[x]+4;
  for y from 1 to min(x-1,floor(sqrt(N^2-x^2))) do
     z:= x^2+y^2;
     if issqr(z) then
       t:= sqrt(z);
       A[t]:= A[t]+8;
     fi
  od
od:
seq(A[i],i=0..N); # Robert Israel, May 08 2015

Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)

a(n)=if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, 2*f[i,2]+1, 1)) \\ Charles R Greathouse IV, Feb 01 2017

a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017

from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 4*r if n > 0 else 0
# Orson R. L. Peters, Jan 31 2017

a(n) = A000328(n) - A051132(n).
a(n) = 8*A046080(n) + 4 for n > 0.
a(n) = A004018(n^2).
From Jean-Christophe Hervé, Dec 01 2013: (Start)
a(A084647(k)) = 28.
a(A084648(k)) = 36.
a(A084649(k)) = 44. (End)
a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017
a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018
From Hugo Pfoertner, Sep 21 2023: (Start)
a(n) = 8*A063014(n) - 4 for n > 0.
a(n) = 4*A256452(n) for n > 0. (End)

A057655 The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.

Original entry on oeis.org

1, 5, 9, 9, 13, 21, 21, 21, 25, 29, 37, 37, 37, 45, 45, 45, 49, 57, 61, 61, 69, 69, 69, 69, 69, 81, 89, 89, 89, 97, 97, 97, 101, 101, 109, 109, 113, 121, 121, 121, 129, 137, 137, 137, 137, 145, 145, 145, 145, 149, 161, 161, 169, 177, 177, 177

Offset: 0

N. J. A. Sloane, Oct 15 2000

			a(0) = 1 (counting origin).
a(1) = 5 since 4 points lie on the circle of radius sqrt(1) + origin.
a(2) = 9 since 4 lattice points lie on the circle w/radius = sqrt(2) (along diagonals) + 4 points inside the circle + origin. - _Wesley Ivan Hurt_, Jan 10 2013

C. Alsina and R. B. Nelsen, Charming Proofs: A Journey Into Elegant Mathematics, Math. Assoc. Amer., 2010, p. 42.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
F. Fricker, Einfuehrung in die Gitterpunktlehre, Birkhäuser, Boston, 1982.
P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 5.
E. Kraetzel, Lattice Points, Kluwer, Dordrecht, 1988.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
W. Sierpiński, Elementary Theory of Numbers, Elsevier, North-Holland, 1988.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 245-246.

T. D. Noe, Table of n, a(n) for n = 0..1000
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
F. Richman, Counting Gaussian integers in a disk
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.

Partial sums of A004018. Cf. A014198, A057656, A057961, A057962, A122510. For another version see A000328.

Cf. A038589 (for hexagonal lattice).

a057655 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 <= n]
-- Reinhard Zumkeller, Jan 23 2012

N:= 1000: # to get a(0) to a(N)
R:= Array(0..N):
for a from 0 to floor(sqrt(N)) do
  for b from 0 to floor(sqrt(N-a^2)) do
    r:= a^2 + b^2;
    R[r]:= R[r] + (2 - charfcn[0](a))*(2 - charfcn[0](b));
  od
od:
convert(map(round,Statistics:-CumulativeSum(R)),list); # Robert Israel, Sep 29 2014

f[n_] := 1 + 4Sum[ Floor@ Sqrt[n - k^2], {k, 0, Sqrt[n]}]; Table[ f[n], {n, 0, 60}] (* Robert G. Wilson v, Jun 16 2006 *)
Accumulate[ SquaresR[2, Range[0, 55]]] (* Jean-François Alcover, Feb 24 2012 *)
CoefficientList[Series[EllipticTheta[3,0,x]^2/(1-x), {x, 0, 100}], x] (* Vaclav Kotesovec, Sep 29 2014 after Robert Israel *)

a(n)=sum(x=-n,n,sum(y=-n,n,if((sign(x^2+y^2-n)+1)*sign(x^2+y^2-n),0,1)))

a(n)=1+4*sum(k=0,sqrtint(n), sqrtint(n-k^2) ); /* Benoit Cloitre, Oct 08 2012 */

from math import isqrt
def A057655(n): return 1+(sum(isqrt(n-k**2) for k in range(isqrt(n)+1))<<2) # Chai Wah Wu, Jul 31 2023

a(n) = 1 + 4*{ [n/1] - [n/3] + [n/5] - [n/7] + ... }. - Gauss
a(n) = 1 + 4*Sum_{ k = 0 .. [sqrt(n)] } [ sqrt(n-k^2) ]. - Liouville (?)
a(n) - Pi*n = O(sqrt(n)) (Gauss). a(n) - Pi*n = O(n^c), c = 23/73 + epsilon ~ 0.3151 (Huxley). If a(n) - Pi*n = O(n^c) then c > 1/4 (Landau, Hardy). It is conjectured that a(n) - Pi*n = O(n^(1/4 + epsilon)) for all epsilon > 0.
a(n) = A014198(n) + 1.
a(n) = A122510(2,n). - R. J. Mathar, Apr 21 2010
a(n) = 1 + sum((floor(1/(k+1)) + 4 * floor(cos(Pi * sqrt(k))^2) - 4 * floor(cos(Pi * sqrt(k/2))^2) + 8 * sum((floor(cos(Pi * sqrt(i))^2) * floor(cos(Pi * sqrt(k-i))^2)), i = 1..floor(k/2))), k = 1..n). - Wesley Ivan Hurt, Jan 10 2013
G.f.: theta_3(0,x)^2/(1-x) where theta_3 is a Jacobi theta function. - Robert Israel, Sep 29 2014
a(n^2) = A000328(n). - R. J. Mathar, Aug 03 2025

A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 1, 6, 4, 0, 1, 8, 12, 0, 2, 1, 10, 24, 8, 4, 0, 1, 12, 40, 32, 6, 8, 0, 1, 14, 60, 80, 24, 24, 0, 0, 1, 16, 84, 160, 90, 48, 24, 0, 0, 1, 18, 112, 280, 252, 112, 96, 0, 4, 2, 1, 20, 144, 448, 574, 312, 240, 64, 12, 4, 0, 1, 22, 180, 672, 1136, 840, 544, 320, 24, 30, 8, 0

Offset: 1

R. J. Mathar, Oct 29 2006

This is the transpose of the array in A286815.

T(d,n) is divisible by 2d for any n != 0 iff d is a power of 2. - Jianing Song, Sep 05 2018

			Array T(d,n) with rows d = 1,2,3,... and columns n = 0,1,2,3,... reads
  1  2   0   0    2    0     0     0     0     2      0 ...
  1  4   4   0    4    8     0     0     4     4      8 ...
  1  6  12   8    6   24    24     0    12    30     24 ...
  1  8  24  32   24   48    96    64    24   104    144 ...
  1 10  40  80   90  112   240   320   200   250    560 ...
  1 12  60 160  252  312   544   960  1020   876   1560 ...
  1 14  84 280  574  840  1288  2368  3444  3542   4424 ...
  1 16 112 448 1136 2016  3136  5504  9328 12112  14112 ...
  1 18 144 672 2034 4320  7392 12672 22608 34802  44640 ...
  1 20 180 960 3380 8424 16320 28800 52020 88660 129064 ...

Alois P. Heinz, Antidiagonals d = 1..141, flattened
Wikipedia, Sum of squares function (transposed)
Index entries for sequences related to sums of squares

Cf. A066535, A286815.
Cf. A000122 (1st row), A004018 (2nd row), A005875 (3rd row), A000118 (4th row), A000132 (5th row), A000141 (6th row), A008451 (7th row), A000143 (8th row), A008452 (9th row), A000144 (10th row), A008453 (11th row), A000145 (12th row), A276285 (13th row), A276286 (14th row), A276287 (15th row), A000152 (16th row).
Cf. A005843 (2nd column), A046092 (3rd column), A130809 (4th column).
Cf. A010052 (1st row divides 2), A002654 (2nd row divides 4), A046897 (4th row divides 8), A008457 (8th row divides 16), A302855 (16th row divides 32), A302857 (32nd row divides 64).

A122141 := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+procname(d-1,n-i^2) ; elif n-i^2 = 0 then cnts := cnts+1 ; fi ; fi ; od ; cnts ;
end:
for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",A122141(d,n)) ; od ; od;
# second Maple program:
A:= proc(d, n) option remember; `if`(n=0, 1, `if`(n<0 or d<1, 0,
      A(d-1, n) +2*add(A(d-1, n-j^2), j=1..isqrt(n))))
    end:
seq(seq(A(h-n, n), n=0..h-1), h=1..14); # Alois P. Heinz, Jul 16 2014

Table[ SquaresR[d - n, n], {d, 1, 12}, {n, 0, d - 1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)
A[d_, n_] := A[d, n] = If[n==0, 1, If[n<0 || d<1, 0, A[d-1, n] + 2*Sum[A[d-1, n-j^2], {j, 1, Sqrt[n]}]]]; Table[A[h-n, n], {h, 1, 14}, {n, 0, h-1}] // Flatten (* Jean-François Alcover, Feb 28 2018, after Alois P. Heinz *)

from sympy.core.power import isqrt
from functools import cache
@cache
def T(d, n):
  if n == 0: return 1
  if n < 0 or d < 1: return 0
  return T(d-1, n) + sum(T(d-1, n-(j**2)) for j in range(1, isqrt(n)+1)) * 2  # Darío Clavijo, Feb 06 2024

T(n,n) = A066535(n). - Alois P. Heinz, Jul 16 2014

A286815 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2j))^5/((1 - x^j)(1 - x^(4*j)))^2)^k.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, May 27 2017

A(n,k) is the number of ways of writing n as a sum of k squares.

This is the transpose of the array in A122141.

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 2, 4,  6,  8, ...
   0, 0, 4, 12, 24, ...
   0, 0, 0,  8, 32, ...
   0, 2, 4,  6, 24, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Sum of squares function

Columns k=0-16 give: A000007, A000122, A004018, A005875, A000118, A000132, A000141, A008451, A000143, A008452, A000144, A008453, A000145, A276285, A276286, A276287, A000152.
Diagonal gives A066535.
Cf. A122141.

A:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      A(n, k-1) +2*add(A(n-j^2, k-1), j=1..isqrt(n))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 27 2017

A[n_, k_] := A[n, k] = If[n == 0, 1, If[n < 0 || k < 1, 0, A[n, k-1] + 2*Sum[A[n-j^2, k-1], {j, 1, Sqrt[n]}]]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 28 2018, after Alois P. Heinz *)

G.f. of column k: (Product_{j>=1} (1 - x^(2*j))^5/((1 - x^j)*(1 - x^(4*j)))^2)^k.

A066535 Number of ways of writing n as a sum of n squares.

Original entry on oeis.org

1, 2, 4, 8, 24, 112, 544, 2368, 9328, 34802, 129064, 491768, 1938336, 7801744, 31553344, 127083328, 509145568, 2035437440, 8148505828, 32728127192, 131880275664, 532597541344, 2153312518240, 8710505815360, 35250721087168, 142743029326162, 578472382307304

Offset: 0

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

nonn

			There are a(3) = 8 solutions (x,y,z) of 3 = x^2 + y^2 + z^2: (1,1,1), (-1,-1,-1), 3 permutations of (1,1,-1) and 3 permutations of (1,-1,-1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
John Holley-Reid and Jeremy Rouse, The number of representations of n as a growing number of squares, arXiv:1910.01001 [math.NT], 2019.

Cf. A004018, A005875, A000118, A066536.
Cf. A122141, A166952. - Paul D. Hanna, Oct 25 2009
a(n^2) gives A361431.

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 16 2014

Join[{1}, Table[SquaresR[n, n], {n, 24}]]

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n)); polcoeff(THETA3^n, n)} /* Paul D. Hanna, Oct 25 2009 */

a(n) equals the coefficient of x^n in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 25 2009

a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.2820942036723951157919967... . - Vaclav Kotesovec, Sep 12 2014

Edited by Dean Hickerson, Jan 12 2002
a(0) added by Paul D. Hanna, Oct 25 2009
Edited by R. J. Mathar, Oct 29 2009

A000143 Number of ways of writing n as a sum of 8 squares.

Original entry on oeis.org

1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736, 550368, 693504, 859952

Offset: 0

N. J. A. Sloane

The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - Wolfdieter Lang, Dec 08 2016

			1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).
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, pp. 314 - 315.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.
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.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).
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.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

8th column of A286815. - Seiichi Manyama, May 27 2017
Row d=8 of A122141.
Cf. A008457, A035016, A010815.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

# JacobiTheta3 is defined in A000122.
A000143List(len) = JacobiTheta3(len, 8)
A000143List(37) |> println # Peter Luschny, Mar 12 2018

(sum(x^(m^2),m=-10..10))^8;
with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n),n=0..100)]; # N. J. A. Sloane, Sep 15 2018

Table[SquaresR[8, n], {n, 0, 33}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[8,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* Michael Somos, Sep 25 2005 */

from math import prod
from sympy import factorint
def A000143(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items())<<4 if n else 1 # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*8)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0

Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008

Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005

G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005

a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - Michael Somos, Sep 25 2005

G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005

A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], R. J. Mathar, Jul 02 2012

a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 /24. - Vaclav Kotesovec, Jul 12 2024

A002652 Theta series of Kleinian lattice Z[(1 + sqrt(-7))/ 2] in 1 complex (or 2 real) dimensions.

Original entry on oeis.org

1, 2, 4, 0, 6, 0, 0, 2, 8, 2, 0, 4, 0, 0, 4, 0, 10, 0, 4, 0, 0, 0, 8, 4, 0, 2, 0, 0, 6, 4, 0, 0, 12, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 12, 0, 8, 0, 0, 2, 4, 0, 0, 4, 0, 0, 8, 0, 8, 0, 0, 0, 0, 2, 14, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 12, 0, 0, 0, 0, 0, 4, 4, 6, 0

Offset: 0

N. J. A. Sloane

In other words, theta series of lattice with Gram matrix [2, 1; 1, 4].
The number of integer solutions (x, y) to x^2 + x*y + 2*y^2 = n.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^2 + 6*x^4 + 2*x^7 + 8*x^8 + 2*x^9 + 4*x^11 + 4*x^14 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 4] = 1 + 2*q^2 + 4*q^4 + 6*q^8 + 2*q^14 + 8*q^16 + 2*q^18 + 4*q^22 + 4*q^28 + 10*q^32 + 4*q^36 + 8*q^44 + 4*q^46 + 2*q^50 + 6*q^56 + 4*q^58 + 12*q^64 + 6*q^72 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 467, Entry 5(i).

John Cannon, Table of n, a(n) for n = 0..5000
J. H. Conway and N. J. A. Sloane, Complex and integral laminated lattices, Trans. Amer. Math. Soc., 280 (1983), 463-490.
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. A035182, A133827, A028951.
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), this sequence (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(14), 1), 85); A[1] + 2*A[2] + 4*A[3] + 6*A[5]; /* Michael Somos, Jun 10 2015 */

f[d_] := KroneckerSymbol[-7, d]; a[n_] := 2*Total[f /@ Divisors[n]]; a[0]=1; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9]]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(t2, t3); if( n<1, n==0, t2 = 2 * sum( n=1, (sqrtint( max(0, 4*n - 7)) + 1)\2, x^(n*n - n)); t3 = 1 + 2 * sum( n=1, sqrtint(n), x^(n*n)); polcoeff( t3 * subst(t3, x, x^7) + x^2 * t2 * subst(t2, x, x^7), n))};

{a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n*4\7), 2 * issquare(t = 4*n - 7*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */

{a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1 * A2), n))}; /* Michael Somos, May 28 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 1; 1, 4], n, 1)[n])}; /* Michael Somos, Jun 03 2005 */

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -7, d)))}; /* Michael Somos, Oct 07 2005 */

G.f.: theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7).
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-7, k) * x^k / (1 - x^k). - Michael Somos, Mar 17 2012
Expansion of phi(x) * phi(x^7) + 4 * x^2 * psi(x^2) * psi(x^14) = phi(-x) * phi(-x^7) + 4 * x * psi(x) * psi(x^7) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 17 2012
Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) / (eta(q) * eta(q^2) * eta(q^7) * eta(q^14)) in powers of q. - Michael Somos, May 28 2005
Moebius transform is period 7 sequence [ 2, 2, -2, 2, -2, -2, 0, ...]. - Michael Somos, Oct 07 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5 * v^2 + 4 * w^2 + 2 * u*w - 4 * u*v - 8 * v*w. - Michael Somos, Sep 20 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 + 2*u2^3*u3 + 18*u1*u3*u6^2 + 18*u2*u3^2*u6 + 6*u1*u2^2*u6 + 3*u1^2*u2*u3 - 3*u2*u3^3 - 18*u2*u3*u6^2 - 6*u1*u6^3 - 9*u1*u3^2*u6 - 6*u1*u2^2*u3 - 6*u1^2*u2*u6. - Michael Somos, Jun 03 2005
From Michael Somos, Mar 17 2012: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = 2 * A035182(n) unless n = 0. a(7*n + 5) = a(7*n + 6) = a(9*n + 3) = a(9*n + 6) = 0. a(2*n + 1) = 2 * A133827(n). a(9*n) = a(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(7^e) = 1, b(p^e) = e + 1 if p == 1, 2, 4 (mod 7), b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, Jun 10 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(7) = 2.3748208... . - Amiram Eldar, Dec 16 2023

A004015 Theta series of face-centered cubic (f.c.c.) lattice.

Original entry on oeis.org

1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24, 48, 24, 48, 8, 84, 24, 96, 48, 24, 0, 96, 6, 96, 48, 48, 36, 120, 24, 48, 24, 48, 48, 120, 24, 120, 0, 96, 24, 108, 30, 48, 72, 72, 32, 144, 0, 96, 72, 72, 48, 120, 0, 144, 12, 48, 48, 168, 48, 96

Offset: 0

N. J. A. Sloane

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. = 1 + 12*x + 6*x^2 + 24*x^3 + 12*x^4 + 24*x^5 + 8*x^6 + 48*x^7 + 6*x^8 + ...
G.f. = 1 + 12*q^2 + 6*q^4 + 24*q^6 + 12*q^8 + 24*q^10 + 8*q^12 + 48*q^14 + 6*q^16 + ...
From _Michael Somos_, Jan 05 2012: (Start)
a(2) = 6 since (1, -1, -1) is a solution to x^2 + y^2 + z^2 + x*y + x*z + y*z = 2 and the other 5 solutions are permutations and negations of this one.
a(2) = 6 since (1, 1, -1, -1) is a solution to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 4 and the other 5 solutions are permutations of this one. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 113.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 263.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Charles L. Hohn, a(0) to a(15), animated
G. Nebe and N. J. A. Sloane, Home page for this lattice
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Theta Series
Index entries for sequences related to f.c.c. lattice

Cf. A004013, A005875, A005901, A045828. A055039 gives the positions of the 0's in this sequence.

Cf. A000007, A000122, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_4, ...)

L := Lattice("A",3); A := ThetaSeries(L, 140); A; /* Michael Somos, Nov 13 2014 */

A := Basis( ModularForms( Gamma1(8), 3/2), 70); A[1] + 12*A[2] + 6*A[3] + 24*A[4]; /* Michael Somos, Sep 08 2018 */

maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a,q,maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a,q,maxd); th4 := series(subs(q=-q, th3),q,maxd); series((1/2)*(th3^3+th4^3),q,200);

a[n_] := SquaresR[3, 2n]; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 + EllipticTheta[ 4, 0, q]^3) / 2, {q, 0, 2 n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^3 + 12 q QPochhammer[ q^4]^3 QPochhammer[ q^8]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
SquaresR[3,2*Range[0,70]] (* Harvey P. Dale, Jun 01 2015 *)

{a(n) = if( n<0, 0, n*=2; polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, Oct 25 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / eta(x^2 + A)^2 / eta(x^8 + A)^2)^3 + 12 * x * eta(x^4 + A)^3 * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))}; /* Michael Somos, May 17 2008 */

{a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 1; 1, 2, 1; 1, 1, 2], n, 1)[n])}; /* Michael Somos, Jan 02 2012 */

from math import prod, isqrt
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1
def A004015(n): return A004018(m:=n<<1)+(sum(A004018(m-k**2) for k in range(1,isqrt(m)+1))<<1) # Chai Wah Wu, Feb 24 2025

Expansion of phi(q^2)^3 + 12 * q * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
Expansion of (phi(q)^3 + phi(-q)^3) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of b(q) * phi(q^18) + c(q^3) * phi(q^2) in powers of q^3 where b(), c() are cubic AGM theta functions and phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004013.
a(n) = A005875(2*n).
G.f.: Sum_{i, j, k in Z} x^( i*i + j*j + k*k + i*j + i*k + j*k ). - Michael Somos, Jan 02 2012
From Michael Somos, Jan 05 2012: (Start)
Number of integer solutions to x^2 + y^2 + z^2 + x*y + x*z + y*z = n.
Number of integer solutions to x + y + z even and x^2 + y^2 + z^2 = 2 * n.
Number of integer solutions to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 2 * n. (End)
a(2*n) = A005875(n). a(2*n+1) = 12 * A045828(n). - Michael Somos, Dec 28 2017

A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

Original entry on oeis.org

0, 1, 5, 25, 65, 325, 1105, 4225, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

The number of lattice points (i,j) on the circle with i^2 + j^2 = a(n) is given by A071385(n).
In a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points", James R. Buddenhagen asked: Which circles have the property that they pass through more lattice points than any smaller circle? and he gave the terms 1, 25, 65, 325, 1105, 4225, 5525, with the missing 5 added by Ahmed Fares. In the same thread Gerry Myerson mentioned the factorization into primes of the form 4*k+1.
Also, numbers with a record number of divisors all of whose prime factors are of the form 4k + 1. - Amiram Eldar, Sep 12 2019
Indices of records of A004018. Apart from the first term, also indices of records of A002654. - Jianing Song, May 20 2021

Ray Chandler, Table of n, a(n) for n = 1..425 (first 97 terms from Ray Chandler, terms 98-365 from Amiram Eldar)
James Buddenhagen, Circle with 3 lattice points, thread in sci.math (May 2002)
Hugo Pfoertner, Construction of the sequences A071383, A071384, A071385

Cf. A000448, A048610, A052199, A071384, A071385, A230655, A300162. Subsequence of A054994 (excluding first term). Where records occur in A004018. See A088959 for circles with integer radius.

Indices of records of Sum_{d|n} kronecker(m, d): A230655 (m=-3), this sequence (m=-4), A279541 (m=-6).

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

from math import prod
from sympy import isprime
primes_congruent_1_mod_4 = [5]
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
def generate_A054994():
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(r,exponents):
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0:
        yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
n,record,radius=1,1,0
print(radius, end="") # or record, for A071385
for radius,expo in generate_A054994():
    num_points = 4*prod((e+1) for e in expo)
    if num_points>record:
        record = num_points
        n += 1
        print (",", radius, end="") # or record, for A071385
        if n==27: break
print()
# Günter Rote, Sep 12 2023

For n>1 we have 1 < a(n+1)/a(n) <= 5, since one can multiply the points x+iy for which x^2 + y^2 = N by either 2+i or 2-i to get two new sets of points X+iY for which X^2 + Y^2 = 5N. This strictly increases the number since it is easy to see that the two sets aren't the same. - J. H. Conway, Jun 04 2002

lim n ->infinity Log(a(n))/n = 1. [Conjectured by Benoit Cloitre, proved by J. H. Conway]

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A063725 Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n.

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A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

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A057655 The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.

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A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read by ascending antidiagonals.

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A286815 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2*j))^5/((1 - x^j)*(1 - x^(4*j)))^2)^k.

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A066535 Number of ways of writing n as a sum of n squares.

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A286815 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2j))^5/((1 - x^j)(1 - x^(4*j)))^2)^k.