A045853 Number of nonnegative solutions of x1^2 + x2^2 + ... + x12^2 = n.

1, 12, 66, 220, 507, 924, 1584, 2772, 4521, 6436, 8712, 12552, 18041, 23364, 28776, 37896, 50997, 62832, 72996, 89892, 115776, 139348, 156816, 185064, 231759, 274044, 300828, 343564, 418638, 487080, 528528, 592284, 707421, 814836, 874170, 959508, 1128338

Offset

0,2

Links

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

Formula

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^12.

G.f.: ((1 + theta_3(x)) / 2)^12. - Ilya Gutkovskiy, Feb 10 2021

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, 12):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2021

Mathematica

CoefficientList[((1 + EllipticTheta[3, 0, q])/2)^12 + O[q]^40, q] (* Jean-François Alcover, Mar 01 2021 *)

Prog

(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(12, 100) # Seiichi Manyama, May 28 2017

Crossrefs

Cf. A010052, A045847.

Sequence in context: A080559 A284641 A226235 * A277104 A014787 A007249

Adjacent sequences: A045850 A045851 A045852 * A045854 A045855 A045856

Keyword

nonn

Author

N. J. A. Sloane

Status

approved