A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

1, 5, 11, 15, 20, 32, 44, 48, 54, 70, 88, 100, 108, 124, 148, 160, 165, 189, 219, 235, 253, 281, 305, 317, 329, 357, 399, 427, 439, 475, 523, 539, 545, 581, 623, 659, 688, 716, 764, 792, 810, 858, 918, 946, 970, 1030, 1078, 1102, 1110, 1154, 1226, 1274, 1304, 1352

Offset

0,2

Links

Table of n, a(n) for n=0..53.

Formula

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^4. - Ilya Gutkovskiy, Mar 14 2017

Mathematica

nn = 50; t = Table[0, {nn}]; Do[d = x^2 + y^2 + z^2 + u^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}, {z, 0, nn}, {u, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

Prog

(Python)
for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        for z in range(99):
            sz = sy + z*z
            if sz>n: break
            for u in range(99):
              su = sz + u*u
              if su>n: break
              k+=1
  print(str(k), end=', ')

Crossrefs

Cf. A014110 (first differences).

Cf. A224212 (number of nonnegative solutions to x^2 + y^2 <= n).

Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).

Cf. A046895 (number of integer solutions to x^2 + y^2 + z^2 + u^2 <= n).

Sequence in context: A314035 A314036 A314037 * A036787 A182664 A299976

Adjacent sequences: A224210 A224211 A224212 * A224214 A224215 A224216

Keyword

nonn

Author

Alex Ratushnyak, Apr 01 2013

Status

approved