A045854 Number of nonnegative solutions of x1^2 + x2^2 + ... + x24^2 = n.

1, 24, 276, 2024, 10650, 43056, 140668, 388608, 948267, 2121176, 4448292, 8811024, 16535160, 29632464, 51256788, 86069680, 140300001, 222302544, 344353516, 523941288, 782700672, 1146771168, 1653111384, 2354351232, 3312339849, 4594531176, 6293753580, 8546252072

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)

Crossrefs

Cf. A010052, A045847.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= b(n, 24):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2021

Mathematica

CoefficientList[((1 + EllipticTheta[3, 0, q])/2)^24 + O[q]^40, q] (* Jean-François Alcover, Mar 01 2021 *)

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(24, 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 + ...)^24.

G.f.: ((1 + theta_3(x)) / 2)^24. - Ilya Gutkovskiy, Feb 10 2021