A045849 Number of nonnegative solutions of x1^2 + x2^2 + ... + x7^2 = n.

1, 7, 21, 35, 42, 63, 112, 141, 126, 154, 259, 315, 280, 308, 462, 567, 497, 462, 693, 910, 798, 749, 1078, 1281, 1092, 1043, 1407, 1715, 1576, 1449, 1946, 2422, 2016, 1687, 2429, 3045, 2604, 2345, 3066

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

Cf. A010052, A038671, A045847.

Programs

Mathematica

(1 + EllipticTheta[3, 0, q])^7/128 + O[q]^50 // CoefficientList[#, q]& (* Jean-François Alcover, Aug 26 2019 *)

PARI

seq(n)=Vec((sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))^7) \\ 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(7, 100) # Seiichi Manyama, May 28 2017

Formula

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^7.

G.f.: (1 + theta_3(q))^7/128, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018