A045854 - OEIS

%I #24 Mar 01 2021 06:10:57

%S 1,24,276,2024,10650,43056,140668,388608,948267,2121176,4448292,

%T 8811024,16535160,29632464,51256788,86069680,140300001,222302544,

%U 344353516,523941288,782700672,1146771168,1653111384,2354351232,3312339849,4594531176,6293753580,8546252072

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

%H Seiichi Manyama, <a href="/A045854/b045854.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from T. D. Noe)

%F Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^24.

%F G.f.: ((1 + theta_3(x)) / 2)^24. - _Ilya Gutkovskiy_, Feb 10 2021

%p b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,

%p       b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))

%p     end:

%p a:= b(n, 24):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 10 2021

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

%o (Ruby)

%o def mul(f_ary, b_ary, m)

%o   s1, s2 = f_ary.size, b_ary.size

%o   ary = Array.new(s1 + s2 - 1, 0)

%o   (0..s1 - 1).each{|i|

%o     (0..s2 - 1).each{|j|

%o       ary[i + j] += f_ary[i] * b_ary[j]

%o     }

%o   }

%o   ary[0..m]

%o end

%o def power(ary, n, m)

%o   if n == 0

%o     a = Array.new(m + 1, 0)

%o     a[0] = 1

%o     return a

%o   end

%o   k = power(ary, n >> 1, m)

%o   k = mul(k, k, m)

%o   return k if n & 1 == 0

%o   return mul(k, ary, m)

%o end

%o def A(k, n)

%o   ary = Array.new(n + 1, 0)

%o   (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}

%o   power(ary, k, n)

%o end

%o p A(24, 100) # _Seiichi Manyama_, May 28 2017

%Y Cf. A010052, A045847.

%K nonn

%O 0,2

%A _N. J. A. Sloane_