A181787 -id:A181787 - 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]

A181788 Number of solutions to n^2 = a^2 + b^2 + c^2 with nonnegative a, b, c.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 6, 9, 3, 15, 9, 12, 6, 15, 9, 24, 3, 18, 15, 18, 9, 36, 12, 21, 6, 27, 15, 42, 9, 27, 24, 27, 3, 51, 18, 39, 15, 33, 18, 54, 9, 36, 36, 36, 12, 69, 21, 39, 6, 51, 27, 69, 15, 45, 42, 54, 9, 81, 27, 48, 24, 51, 27, 117, 3, 63, 51, 54, 18, 96, 39, 57, 15, 60, 33, 102, 18, 90, 54, 63, 9, 123, 36, 66, 36, 78, 36, 114, 12, 72, 69, 93, 21, 126, 39, 84, 6, 78, 51, 168, 27

Offset: 0

T. D. Noe, Nov 12 2010

nonn

Paul D. Hanna and Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A016725, A016727, A181786, A181787.

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,0,nn}, {b,0,nn}, {c,0,nn}]; Prepend[t,1]

{a(n)=local(G=sum(k=0,n,x^(k^2)+x*O(x^(n^2))));polcoeff(G^3,n^2)} /* Paul D. Hanna */

A(n)=my(G=sum(k=0,n,x^(k^2),x*O(x^(n^2)))^3); vector(n+1, k, polcoeff(G,(k-1)^2)) \\ Charles R Greathouse IV, Apr 20 2012

G.f.: [x^(n^2)] G(x)^3 where G(x) = Sum_{k>=0} x^(k^2); the notation [x^(n^2)] G(x)^3 denotes the coefficient of x^(n^2) in G(x)^3. [From Paul D. Hanna, Apr 20 2012]

A212091 Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x^2+y^2+z^2.

Original entry on oeis.org

0, 0, 0, 3, 3, 3, 6, 12, 12, 24, 24, 33, 36, 42, 48, 63, 63, 72, 84, 99, 99, 132, 141, 159, 162, 174, 180, 219, 225, 243, 258, 282, 282, 330, 339, 369, 381, 405, 420, 465, 465, 492, 525, 558, 567, 627, 645, 681, 684, 732, 744, 804, 810, 846, 885, 930

Offset: 0

Clark Kimberling, May 02 2012

nonn

Every term is divisible by 3. For a guide to related sequences, see A211795.

Cf. A211795.

Partial sums of A181787.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 == x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212091 *)
%/3  (* integers *)
(* Peter J. C. Moses, Apr 13 2012 *)

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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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A181788 Number of solutions to n^2 = a^2 + b^2 + c^2 with nonnegative a, b, c.

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A212091 Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x^2+y^2+z^2.

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