A000132 Number of ways of writing n as a sum of 5 squares.

1, 10, 40, 80, 90, 112, 240, 320, 200, 250, 560, 560, 400, 560, 800, 960, 730, 480, 1240, 1520, 752, 1120, 1840, 1600, 1200, 1210, 2000, 2240, 1600, 1680, 2720, 3200, 1480, 1440, 3680, 3040, 2250, 2800, 3280, 4160, 2800, 1920, 4320, 5040, 2800, 3472, 5920

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The units digit of a(n) is 2 if n=5*t^2 for some natural number t, and 0 otherwise. See Moreno & Wagstaff, p. 258, exercise 2. - Ant King, Mar 17 2013

See A025429 for the number of partitions of n into five nonzero squares. - M. F. Hasler, May 30 2014

Also, theta series of lattice Z^5. - Sean A. Irvine, Jul 27 2020

Examples

			G.f. = 1 + 10*x + 40*x^2 + 80*x^3 + 90*x^4 + 112*x^5 + 240*x^6 + ...
		

References

• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 128.
• J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006). [Ant King, Mar 17 2013]

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
• S. Cooper, Sums of five, seven and nine squares, Ramanujan J., vol 6, no. 4, (2002) 469-490.
• 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

Crossrefs

5th column of A286815. - Seiichi Manyama, May 27 2017

Row d=5 of A122141.

Cf. A000118, A025429.

Programs

Mathematica

Table[SquaresR[5, n], {n, 0, 46}] (* Ray Chandler, Nov 28 2006 *)
SquaresR[5,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)

PARI

a(n, k=5) = if(n==0, return(1)); if(k <= 0, return(0)); if(k==1, return(issquare(n))); my(count = 0); for(v = 0, sqrtint(n), count += (2 - (v == 0))*if(k > 2, a(n - v^2, k-1), issquare(n - v^2) * (2 - (n - v^2 == 0)))); count; \\ Daniel Suteu, Aug 28 2021

Python

# uses Python code from A000118
from math import isqrt
def A000132(n): return A000118(n)+(sum(A000118(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

Sage

Q = DiagonalQuadraticForm(ZZ, [1]*5)
Q.representation_number_list(47) # Peter Luschny, Jun 20 2014

Formula

G.f.: (Sum_{j=-oo..+oo} x^(j^2))^5. - R. J. Mathar, Jul 31 2007

a(n) = (10/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

a(n) = A000118(n) + 2*Sum_{k=1..floor(sqrt(n))} A000118(n - k^2). - Daniel Suteu, Aug 28 2021

Extensions

Extended by Ray Chandler, Nov 28 2006