A181788 -id:A181788 - cp's OEIS frontend

A181786 Number of inequivalent solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 3, 0, 2, 3, 3, 0, 6, 2, 3, 1, 2, 1, 8, 1, 3, 3, 4, 0, 10, 2, 5, 3, 4, 3, 8, 0, 5, 6, 6, 2, 11, 3, 6, 1, 8, 2, 12, 1, 6, 8, 8, 1, 15, 3, 8, 3, 7, 4, 20, 0, 6, 10, 9, 2, 16, 5, 9, 3, 9, 4, 15, 3, 15, 8, 10, 0, 22, 5, 11, 6, 9, 6, 18, 2, 11, 11, 14, 3, 21, 6, 13, 1, 12, 8, 31, 2

Offset: 0

T. D. Noe, Nov 12 2010

nonn

Note that a(n)=0 for n=0 and the n in A094958.
Also note that a(2n)=a(n), e.g., a(1000)=a(500)=a(250)=a(125)=14. - Zak Seidov, Mar 02 2012
a(n) is the number of distinct parallelepipeds each one having integer diagonal n and integer sides. - César Eliud Lozada, Oct 26 2014

Samuel Harkness, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)

Cf. A016725, A016727, A046080, A181787, A181788

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,nn}, {b,a,nn}, {c,b,nn}]; Prepend[t,0]

A181787 Number of solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 3, 6, 0, 12, 0, 9, 3, 6, 6, 15, 0, 9, 12, 15, 0, 33, 9, 18, 3, 12, 6, 39, 6, 18, 15, 24, 0, 48, 9, 30, 12, 24, 15, 45, 0, 27, 33, 33, 9, 60, 18, 36, 3, 48, 12, 60, 6, 36, 39, 45, 6, 78, 18, 45, 15, 42, 24, 114, 0, 36, 48, 51, 9, 93, 30, 54, 12, 51, 24, 87, 15, 87, 45, 60, 0, 120, 27, 63, 33, 51, 33, 105, 9, 63, 60, 84, 18, 123, 36, 75, 3, 69, 48, 165, 12

Offset: 0

T. D. Noe, Nov 12 2010

Note that a(n)=0 for n=0 and the n in A094958.

			a(3)=3 because 3^2 = 1^2+2^2+2^2 = 2^2+1^2+2^2 = 2^2+2^2+1^2. - _Robert Israel_, Aug 02 2019

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A016725, A016727, A181786, A181788.

N:= 200: # for a(0)..a(N)
A:= Array(0..N):
mults:= [1,3,6]:
for a from 1 while 3*a^2 <= N^2 do
  if a::odd then b0:= a+1; db:= 2 else b0:= a; db:= 1 fi;
  for b from b0 by db while a^2 + 2*b^2 <= N^2 do
    if (a+b)::odd then c0:= b + (b mod 2); dc:= 2 else c0:= b; dc:= 1 fi;
    for c from c0 by dc do
      v:= a^2 + b^2 + c^2;
      if v > N^2 then break fi;
      if issqr(v) then
        w:= sqrt(v);
        A[w]:= A[w]+ mults[nops({a,b,c})];
      fi
od od od:
convert(A,list); # Robert Israel, Aug 02 2019

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,nn}, {b,nn}, {c,nn}]; Prepend[t,0]

a(n) = A063691(n^2). - Michel Marcus, Apr 25 2015

a(2*n) = a(n). - Robert Israel, Aug 02 2019

A218711 Number of nonnegative solutions to x^2 + y^2 + z^2 < n^2.

Original entry on oeis.org

0, 1, 8, 23, 51, 90, 157, 230, 341, 471, 639, 835, 1063, 1340, 1671, 2022, 2443, 2893, 3428, 4004, 4653, 5359, 6133, 6977, 7907, 8886, 9991, 11152, 12428, 13724, 15192, 16683, 18358, 20072, 21932, 23880, 25941, 28117, 30397, 32822, 35376, 38013, 40840, 43765, 46880, 50090, 53448, 56911, 60583, 64379

Offset: 0

Jon Perry, Nov 04 2012

nonn

Cf. A000604, A181788.

for (i=0;i<50;i++) {
d=0;
for (a=0;a<=i;a++)
for (b=0;b<=i;b++)
for (c=0;c<=i;c++)
if (Math.pow(a,2)+Math.pow(b,2)+Math.pow(c,2)

a(n) = A000604(n) - A181788(n).

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A181786 Number of inequivalent solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

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A181787 Number of solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

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A218711 Number of nonnegative solutions to x^2 + y^2 + z^2 < n^2.

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