OFFSET
0,5
COMMENTS
Number of ways of writing k as a sum of n squares.
REFERENCES
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).
LINKS
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.
EXAMPLE
[ 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
MAPLE
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
MATHEMATICA
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 *)
PROG
(Sage)
for n in (0..10):
Q = DiagonalQuadraticForm(ZZ, [1]*n)
print(Q.theta_series(10).list())
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 01 2018
STATUS
approved