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Number of inequivalent solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.
+10
7
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
COMMENTS
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
MATHEMATICA
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]
Number of solutions to n^2 = a^2 + b^2 + c^2 with nonnegative a, b, c.
+10
4
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
FORMULA
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]
MATHEMATICA
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]
PROG
(PARI) {a(n)=local(G=sum(k=0, n, x^(k^2)+x*O(x^(n^2)))); polcoeff(G^3, n^2)} /* Paul D. Hanna */ (PARI) 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
Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x^2+y^2+z^2.
+10
3
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
COMMENTS
Every term is divisible by 3. For a guide to related sequences, see A211795.
MATHEMATICA
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 *)
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