A045850 Number of nonnegative solutions of x1^2 + x2^2 + ... + x8^2 = n.

1, 8, 28, 56, 78, 112, 196, 288, 309, 344, 532, 736, 756, 784, 1092, 1456, 1486, 1400, 1792, 2464, 2562, 2352, 2940, 3872, 3864, 3536, 4256, 5432, 5608, 5216, 6076, 7840, 7933, 6832, 7952, 10472, 10494

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)

Index entries for sequences related to sums of squares

Formula

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^8.

G.f.: (1 + theta_3(q))^8/256, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

Mathematica

(1 + EllipticTheta[3, 0, q])^8/256 + O[q]^40 // CoefficientList[#, q]& (* Jean-François Alcover, Aug 26 2019 *)

Prog

(PARI) seq(n)=Vec((sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))^8) \\ Andrew Howroyd, Aug 08 2018
(Ruby)
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def power(ary, n, m)
  if n == 0
    a = Array.new(m + 1, 0)
    a[0] = 1
    return a
  end
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A(k, n)
  ary = Array.new(n + 1, 0)
  (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
  power(ary, k, n)
end
p A(8, 100) # Seiichi Manyama, May 28 2017

Crossrefs

Cf. A010052, A045847.

Sequence in context: A230211 A229393 A173681 * A264354 A033580 A299289

Adjacent sequences: A045847 A045848 A045849 * A045851 A045852 A045853

Keyword

nonn

Author

N. J. A. Sloane

Status

approved