A045847 -id:A045847 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

Cf. A010052, A045847.

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

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

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

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

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

Original entry on oeis.org

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

N. J. A. Sloane

nonn

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

Cf. A010052, A045847.

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

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

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

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

cp's OEIS Frontend

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

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