A302996 -id:A302996 - cp's OEIS frontend

A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 29, 17, 6, 1, 1, 7, 36, 70, 54, 26, 7, 1, 1, 8, 63, 157, 165, 99, 35, 8, 1, 1, 9, 106, 337, 482, 357, 163, 45, 9, 1, 1, 10, 171, 702, 1319, 1203, 688, 239, 58, 10, 1, 1, 11, 265, 1420, 3390, 3819, 2673, 1154, 344, 73, 11, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_k)^2 <= n^2.

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  1,  2,   3,   4,    5,     6,  ...
  1,  3,   6,  11,   20,    36,  ...
  1,  4,  11,  29,   70,   157,  ...
  1,  5,  17,  54,  165,   482,  ...
  1,  6,  26,  99,  357,  1203,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409.
Rows n=0..10 give A000012, A000027, A055417, A055418, A055419, A055420, A055421, A055422, A055423, A055424, A055425.
Main diagonal gives A302863.
Cf. A000122, A122510, A302996, A302997.

Table[Function[k, SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^k/(2^k (1 - x)), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, 0, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

T(n,k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k.

A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.

			Square array begins:
  1,   1,   1,    1,     1,      1,  ...
  1,   3,   5,    7,     9,     11,  ...
  1,   5,  13,   33,    89,    221,  ...
  1,   7,  29,  123,   425,   1343,  ...
  1,   9,  49,  257,  1281,   5913,  ...
  1,  11,  81,  515,  3121,  16875,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.
Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.
Main diagonal gives A302861.
Cf. A000122, A122510, A302996, A302998.

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k/(1 - x), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j=-infinity..infinity} x^(j^2))^k.

A232173 Number of ways of writing n^2 as a sum of n squares.

Original entry on oeis.org

1, 2, 4, 30, 24, 1210, 18396, 235998, 4793456, 76168850, 1282320348, 25100418046, 481341997032, 10452086347274, 237925595533164, 5524220670435982, 136705837928870368, 3444192369181374754, 89772662325079950436, 2431910317560215089758, 67517711482300160612104

Offset: 0

Paul D. Hanna, Nov 19 2013

nonn

			There are a(4) = 24 solutions (w,x,y,z) of 4^2 = w^2 + x^2 + y^2 + z^2:
(2,2,2,2), (-2,-2,-2,-2), 6 permutations of (2,2,-2,-2),
4 permutations of (2,2,2,-2), 4 permutations of (2,-2,-2,-2),
4 permutations of (4,0,0,0), and 4 permutations of (-4,0,0,0).
To illustrate a(n) = the coefficient of x^(n^2) in theta_3(x)^n, where
theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + 2*x^49 +...,
form a table of coefficients of x^k in theta_3(x)^n, n>=0, like so:
n\k:0..1...2...3...4...5...6...7...8...9..10..11..12..13..14..15..16....
0:[(1),0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,...];
1: [1,(2), 0,  0,  2,  0,  0,  0,  0,  2,  0,  0,  0,  0,  0,  0,  2,...];
2: [1, 4,  4,  0, (4), 8,  0,  0,  4,  4,  8,  0,  0,  8,  0,  0,  4,...];
3: [1, 6, 12,  8,  6, 24, 24,  0, 12,(30),24, 24,  8, 24, 48,  0,  6,...];
4: [1, 8, 24, 32, 24, 48, 96, 64, 24,104,144, 96, 96,112,192,192,(24),...];
5: [1,10, 40, 80, 90,112,240,320,200,250,560,560,400,560,800,960,730,...];
then the coefficients in parenthesis form the initial terms of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (first 101 terms from Paul D. Hanna)

Cf. A066535.

Main diagonal of A302996.

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, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 10 2023

b[n_, t_] := b[n, t] = If[n == 0, 1, If[n < 0 || t < 1, 0, b[n, t - 1] + 2*Sum[b[n - j^2, t - 1], {j, 1, Floor@Sqrt[n]}]]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 28 2023, after Alois P. Heinz *)

{a(n)=local(THETA3=1+2*sum(m=1, n+1, x^(m^2))+x*O(x^(n^2))); polcoeff(THETA3^n, n^2)}
for(n=0, 30, print1(a(n), ", "))

a(n) equals the coefficient of x^(n^2) in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Feb 17 2015

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 2,  2,  2,  2,  2,  2,  2, ...
  0, 1, 1, 2, 2,  2,  3,  3,  3,  4,  4, ...
  0, 1, 1, 1, 2,  3,  3,  4,  5,  5,  6, ...
  0, 1, 2, 2, 3,  4,  5,  7,  8,  9, 11, ...
  0, 1, 1, 2, 4,  5,  9, 10, 11, 15, 17, ...
  0, 1, 1, 2, 4,  6,  9, 13, 18, 21, 27, ...
  0, 1, 1, 1, 2,  7,  9, 16, 25, 30, 41, ...
  0, 1, 1, 4, 6,  8, 18, 27, 36, 52, 68, ...
  0, 1, 2, 2, 7, 13, 23, 36, 51, 70, 94, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000012, A063014, A016727, A065458, A065459, A065460, A065461, A065462, A255213, A255214.
Main diagonal gives A105152.
Cf. A302996.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1 and n<=t, 1,
      (j-> `if`(t*jn, 0, b(n-j, i, t-1))))(i^2))
    end:
A:= (n, k)-> b(n^2, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1 && n <= t, 1, Function[j, If[t*jn, 0, b[n-j, i, t-1]]]][i^2]]; A[n_, k_] := b[n^2, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A016725 Number of integer solutions to x^2+y^2+z^2 = n^2, allowing zeros and distinguishing signs and order.

Original entry on oeis.org

1, 6, 6, 30, 6, 30, 30, 54, 6, 102, 30, 78, 30, 78, 54, 150, 6, 102, 102, 126, 30, 270, 78, 150, 30, 150, 78, 318, 54, 174, 150, 198, 6, 390, 102, 270, 102, 222, 126, 390, 30, 246, 270, 270, 78, 510, 150, 294, 30, 390, 150, 510, 78, 318, 318, 390, 54, 630, 174, 366

Offset: 0

csvcjld(AT)nomvst.lsumc.edu

Hurwitz found a formula for a(n). See the paper by Olds.

			1 + 6*x + 6*x^2 + 30*x^3 + 6*x^4 + 30*x^5 + 30*x^6 + 54*x^7 + 6*x^8 + ...

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Werner Hürlimann, Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.5.
Jean Lagrange, Décomposition d'un entier en somme de carrés et fonction multiplicative, Séminaire Delange-Pisot-Poitou. Théorie des nombres, 14 no. 1 (1972-1973), Exp. No. 1, 5 p.
C. D. Olds, On the representations, N_3(n^2), Bull. Amer. Math. Soc. 47 (1941), 499-503.
Eric Weisstein's World of Mathematics, Sum of Squares Function

Cf. A005875.

Column k=3 of A302996.

for n from 0 to 60 do s:=0: for x from -n to n do for y from -n to n do for z from -n to n do if (x^2+y^2+z^2) = n^2 then s:=s+1 fi od od od: printf("%d, ",s) od: # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 13 2004

Mathematica
```
SquaresR[3, Range[0,100]^2]
```

{a(n) = if( n<1, n==0, polcoeff( sum( k=1, n, 2 * x^k^2, 1 + x * O(x^n^2))^3, n^2))} /* Michael Somos, Nov 18 2011 */

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 6 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 1, p^e + if( p%4 == 1, 0, 2 * (p^e - 1) / (p - 1))))))} /* Michael Somos, Nov 18 2011 */

a(n) = 6 * b(n) if n>0 where b(n) is multiplicative with b(2^e) = 1, b(p^e) = p^e if p == 1 (mod 4), b(p^e) = p^e + 2 * (p^e - 1) / (p - 1) if p == 3 (mod 4). - Michael Somos, Nov 18 2011
a(n) = A005875(n^2).
a(n) = [x^(n^2)] theta_3(x)^3, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

Revised description from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 13 2004

A267326 Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2).

Original entry on oeis.org

1, 8, 24, 104, 24, 248, 312, 456, 24, 968, 744, 1064, 312, 1464, 1368, 3224, 24, 2456, 2904, 3048, 744, 5928, 3192, 4424, 312, 6248, 4392, 8744, 1368, 6968, 9672, 7944, 24, 13832, 7368, 14136, 2904, 11256, 9144, 19032, 744, 13784, 17784, 15144, 3192

Offset: 0

Christopher Heiling, Jan 13 2016

For all pair of relatively prime numbers k, m this sequence is multiplicative with a factor of 8: a(k*m) = 8*a(k)*a(m). - Christopher Heiling, Apr 02 2017

			For n = 2 the a(n) = 24 solutions of x^2 + y^2 + z^2 + t^2 = 2^2 are:
{x,y,z,t} = {{0,0,0,2};{0,0,0,-2};{0,0,2,0};{0,0,-2,0};{0,2,0,0};{0,-2,0,0};{2,0,0,0};{-2,0,0,0};{1,1,1,1};{1,1,1,-1};{1,1,-1,1};{1,-1,1,1};{-1,1,1,1};{1,1,-1,-1};{1,-1,1,-1};{-1,1,1,-1};{1,-1,-1,1};{-1,1,-1,1};{1,-1,-1,-1};{-1,1,-1,-1};{-1,-1,1,-1};{-1,-1,1,-1};{-1,-1,-1,1};{-1,-1,-1,-1}}.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..150 from Christopher Heiling)

Cf. A000118.
Partial sums of this sequence give A264390.
Column k=4 of A302996.

terms := 42:
(add(q^(m^2), m = -terms..terms))^4:
seq(coeff(%, q, n^2), n = 0..terms); # Peter Bala, Jan 15 2016

a[n_] := SquaresR[4, n^2];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 18 2023 *)

a(n) = A264390(n) - A264390(n-1) for n > 1 and a(1) = A264390(1) = 2*D.
a(n) = 8*sigma(n^2) if n is odd else 24*sigma(m(n^2)), where sigma(n) = A000203(n) and m(n) = A000265(n) is the largest odd divisor of n. - Peter Bala, Jan 15 2016
a(p^(k+1)) = 8*(p^2 *a(p^k)+p+1) for p prime. In particular a(p) = 8*(p^2+p+1). - Christopher Heiling, Apr 02 2017

a(0)=1 prepended by Alois P. Heinz, Mar 10 2023

A361695 Number of ways of writing n^2 as a sum of seven squares.

Original entry on oeis.org

1, 14, 574, 3542, 18494, 43414, 145222, 235998, 591934, 860846, 1779974, 2256422, 4678982, 5195750, 9675918, 10983742, 18942014, 19873966, 35294686, 34670454, 57349894, 59707494, 92513302, 90116222, 149759302, 135668414, 213025750, 209185718, 311753358, 287144326, 450333422

Offset: 0

Alois P. Heinz, Mar 22 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to sums of squares

Column k=7 of A302996.

Cf. A008451, A065461.

a:= n-> coeff((sum(x^(j^2), j=-n..n))^7, x, n^2):
seq(a(n), n=0..30);
# second Maple program:
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, 7):
seq(a(n), n=0..30);

SquaresR[7, Range[0, 30]^2] (* Paolo Xausa, Aug 21 2025 *)

a(n) = [x^(n^2)] (Sum_{j=-n..n} x^(j^2))^7.
a(n) = A008451(n^2).
a(n) = A302996(n,7).

A374493 Number of ways of writing n^2 as a sum of 8 squares.

Original entry on oeis.org

1, 16, 1136, 12112, 74864, 252016, 859952, 1887888, 4793456, 8830096, 17893136, 28366288, 56672048, 77264112, 134040048, 190776112, 306783344, 386279728, 626936816, 752843856, 1179182864, 1429131216, 2014006448, 2368768912, 3628646192, 3937752016, 5485751952

Offset: 0

Seiichi Manyama, Jul 12 2024

nonn

Column k=8 of A302996.

Cf. A000143.

SquaresR[8, Range[0,30]^2] (* Paolo Xausa, Aug 21 2025 *)

a000143(n) = if(n==0, 1, 16*sumdiv(n, d, (-1)^(n+d)*d^3));
a(n) = a000143(n^2);

a(n) = [x^(n^2)] (Sum_{j=-n..n} x^(j^2))^8.
a(n) = A000143(n^2).
a(n) is divisible by 16 for n > 0.

cp's OEIS Frontend

A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.

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A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

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A232173 Number of ways of writing n^2 as a sum of n squares.

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A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A016725 Number of integer solutions to x^2+y^2+z^2 = n^2, allowing zeros and distinguishing signs and order.

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A267326 Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2).

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A361695 Number of ways of writing n^2 as a sum of seven squares.

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