A000143 Number of ways of writing n as a sum of 8 squares.
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
Examples
1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...
References
- 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.
Links
- 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
Crossrefs
Programs
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Julia
# JacobiTheta3 is defined in A000122. A000143List(len) = JacobiTheta3(len, 8) A000143List(37) |> println # Peter Luschny, Mar 12 2018
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Maple
(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
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Mathematica
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 *) -
PARI
{a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))} -
PARI
{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 */ -
Python
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
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SageMath
Q = DiagonalQuadraticForm(ZZ, [1]*8) Q.representation_number_list(60) # Peter Luschny, Jun 20 2014
Formula
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
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
Comments