A008451 -id:A008451 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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())

A173677 Number of ways of writing n as a sum of two nonnegative cubes.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004829, A008451, A010057, A051343, A173676.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

list(n)=my(q='q);Vec(sum(m=0,(n+.5)^(1/3),q^(m^3),O(q^(n+1)))^2) \\ Charles R Greathouse IV, Jun 07 2012

a(n) = Sum_{k=0..n} c(k) * c(n-k), where c = A010057. - Wesley Ivan Hurt, Nov 09 2023

A173676 Number of ways of writing n as a sum of seven nonnegative cubes.

Original entry on oeis.org

1, 7, 21, 35, 35, 21, 7, 1, 7, 42, 105, 140, 105, 42, 7, 0, 21, 105, 210, 210, 105, 21, 0, 0, 35, 140, 210, 147, 77, 105, 140, 105, 77, 112, 105, 77, 210, 420, 420, 210, 63, 42, 21, 105, 420, 630, 420, 105, 7, 7, 0, 140, 420, 420, 161, 105, 211, 210, 105, 126, 210, 105, 105, 420, 637, 462, 210, 182, 147, 42, 217, 630, 672, 420, 420, 427, 210, 42

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^7.

It is known that a(n)>0 if n is even and > 454.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. D. Elkies, Every even number greater than 454 is the sum of seven cubes, arXiv:1009.3983

Cf. A004829, A008451.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173678 Number of ways of writing n as a sum of 4 nonnegative cubes.

Original entry on oeis.org

1, 4, 6, 4, 1, 0, 0, 0, 4, 12, 12, 4, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 4, 4, 0, 4, 12, 12, 4, 0, 1, 0, 0, 12, 24, 12, 0, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 12, 12, 4, 12, 12, 4, 0, 0, 6, 0, 12, 24, 12, 0, 0, 0, 0, 0, 12, 16, 4, 0, 0, 0, 0, 0, 4, 4, 0, 12, 24, 12, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004829, A008451, A010057, A173677, A051343, A173676.
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Without order you get A025448.

A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 7, 35, 0, 42, 35, 0, 105, 28, 21, 140, 49, 105, 105, 106, 210, 84, 182, 210, 217, 287, 105, 420, 378, 126, 497, 392, 420, 532, 350, 630, 714, 434, 546, 980, 742, 609, 980, 896, 1071, 882, 875, 1470, 1239, 1099, 1155, 1722, 1652, 882, 1933, 1995, 1554, 2072, 1505

Offset: 7

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A045849, A063691, A063725, A063730, A340481, A340905, A340915, A340946, A340947.

Column k=7 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (theta_3(x) - 1)^7 / 128, where theta_3() is the Jacobi theta function.

A276285 Number of ways of writing n as a sum of 13 squares.

Original entry on oeis.org

1, 26, 312, 2288, 11466, 41808, 116688, 265408, 535704, 1031914, 1899664, 3214224, 5043376, 7801744, 12066912, 17689152, 24443978, 34039200, 48210760, 64966096, 83323344, 109157152, 145532816, 185245632, 227110416, 284788010, 363737712

Offset: 0

Ilya Gutkovskiy, Aug 27 2016

More generally, the ordinary generating function for the number of ways of writing n as a sum of k squares is theta_3(0, q)^k = 1 + 2*k*q + 2*(k - 1)*k*q^2 + (4/3)*(k - 2)*(k - 1)*k*q^3 + (2/3)*((k - 3)*(k - 2)*(k - 1) + 3)*k*q^4 + (4/15) *(k - 1)*k*(k^3 - 9*k^2 + 26*k - 9)*q^5 + ..., where theta is the Jacobi theta functions.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

13th column of A286815. - Seiichi Manyama, May 27 2017
Row d=13 of A122141.
Cf. Number of ways of writing n as a sum of k squares: A004018 (k = 2), A005875 (k = 3), A000118 (k = 4), A000132 (k = 5), A000141 (k = 6), A008451 (k = 7), A000143 (k = 8), A008452 (k = 9), A000144 (k = 10), A008453 (k = 11), A000145 (k = 12), this sequence (k = 13), A000152 (k = 16).

Mathematica
```
Table[SquaresR[13, n], {n, 0, 26}]
```

G.f.: theta_3(0,q)^13, where theta_3(x,q) is the third Jacobi theta function.

a(n) = (26/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

A340998 Number of partitions of n into 7 distinct nonzero squares.

Original entry on oeis.org

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

Offset: 140

Ilya Gutkovskiy, Feb 02 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A025441, A025442, A025443, A025444, A045849, A224982, A340906, A340988, A340999, A341000, A341001.

Column k=7 of A341040.

A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 15, 99, 379, 953, 1793, 3081, 5449, 8893, 12435, 16859, 24419, 33659, 42115, 53203, 69779, 88273, 106081, 125821, 153541, 187981, 217437, 248741, 298469, 351277, 394691, 446939, 515259, 589307, 657683, 728803, 828259, 939223, 1029159, 1124023, 1260103

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A008451.

Index entries for sequences related to sums of squares

Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate

my(q='q+O('q^(55))); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^7/(1-q)) \\ Joerg Arndt, Jun 21 2024

G.f.: theta_3(x)^7 / (1 - x).

a(n^2) = A055413(n).

cp's OEIS Frontend

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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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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A173677 Number of ways of writing n as a sum of two nonnegative cubes.

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A173676 Number of ways of writing n as a sum of seven nonnegative cubes.

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A173678 Number of ways of writing n as a sum of 4 nonnegative cubes.

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A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

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Maple

Mathematica

Formula

A276285 Number of ways of writing n as a sum of 13 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.