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 A000141 Number of ways of writing n as a sum of 6 squares. 17
 1, 12, 60, 160, 252, 312, 544, 960, 1020, 876, 1560, 2400, 2080, 2040, 3264, 4160, 4092, 3480, 4380, 7200, 6552, 4608, 8160, 10560, 8224, 7812, 10200, 13120, 12480, 10104, 14144, 19200, 16380, 11520, 17400, 24960, 18396, 16440, 24480, 27200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The relevant identity for the o.g.f. is theta_3(x)^6 = 1 + 16*Sum_{j>=1} j^2*x^j/(1 + x^(2*j)) - 4*Sum_{j >=0} (-1)^j*(2*j+1)^2 *x^(2*j+1)/(1 - x^(2*j+1)), See the Hardy-Wright reference, p. 315, first equation. - Wolfdieter Lang, Dec 08 2016 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, p. 314. LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 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. Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, 2013, Preprint submitted to Journal of Geometry and Physics. 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. FORMULA Expansion of theta_3(z)^6. a(n) = 4( Sum_{ d|n, d ==3 mod 4} d^2 - Sum_{ d|n, d ==1 mod 4} d^2 ) + 16( Sum_{ d|n, n/d ==1 mod 4} d^2 - Sum_{ d|n, n/d ==3 mod 4} d^2 ) [Jacobi]. [corrected by Sean A. Irvine, Oct 01 2009] a(n) = 16*A050470(n) - 4*A002173(n). - Michel Marcus, Dec 15 2012 a(n) = (12/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017 MAPLE (sum(x^(m^2), m=-10..10))^6; # Alternative: A000141list := proc(len) series(JacobiTheta3(0, x)^6, x, len+1); seq(coeff(%, x, j), j=0..len-1) end: A000141list(40); # Peter Luschny, Oct 02 2018 MATHEMATICA Table[SquaresR[6, n], {n, 0, 40}] (* Ray Chandler, Dec 06 2006 *) SquaresR[6, Range[0, 50]] (* Harvey P. Dale, Aug 26 2011 *) EllipticTheta[3, 0, z]^6 + O[z]^40 // CoefficientList[#, z]& (* Jean-François Alcover, Dec 05 2019 *) PROG (Haskell) a000141 0 = 1 a000141 n = 16 * a050470 n - 4 * a002173 n -- Reinhard Zumkeller, Jun 17 2013 (Sage) Q = DiagonalQuadraticForm(ZZ, [1]*6) Q.representation_number_list(40) # Peter Luschny, Jun 20 2014 CROSSREFS Row d=6 of A122141 and of A319574, 6th column of A286815. Cf. A050470, A002173. Sequence in context: A153792 A229616 A321465 * A328094 A300758 A332544 Adjacent sequences:  A000138 A000139 A000140 * A000142 A000143 A000144 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Extended by Ray Chandler, Nov 28 2006 STATUS approved

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Last modified August 14 09:14 EDT 2020. Contains 336480 sequences. (Running on oeis4.)