A016727 -id:A016727 - cp's OEIS frontend

A000164 Number of partitions of n into 3 squares (allowing part zero).

1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 0, 2, 1, 0, 1, 2, 2, 1, 2, 1, 2, 0, 1, 3, 1, 1, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 3, 0, 1, 2, 1, 2, 0, 2, 2, 0, 1, 3, 3, 1, 2, 2, 1, 0, 2, 2, 3, 2, 1, 2, 1, 0, 1, 4, 2, 2, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 2, 0, 1, 2, 3, 3, 2, 4, 2, 0, 2

Offset: 0

N. J. A. Sloane

nonn

a(n) = number of triples of integers [ i, j, k] such that i >= j >= k >= 0 and n = i^2 + j^2 + k^2. - Michael Somos, Jun 05 2012

			G.f. = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^8 + 2*x^9 + x^10 + x^11 + x^12 + x^13 + ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.

T. D. Noe, Table of n, a(n) for n = 0..10000
Hirschhorn, M. D., Some formulas for partitions into squares, Discrete Math. 211 (2000), pp. 225-228.

Equivalent sequences for other numbers of squares: A000161 (2), A002635 (4), A000174 (5).
Cf. A004215 (positions of zeros), A094942 (positions of ones), A124966 (positions of greater values).
Cf. A005875, A016727.

A000164 := proc(n)
    local a,x,y,z2,z ;
    a := 0 ;
    for x from 0 do
        if 3*x^2 > n then
            return a;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            end if;
            z2 := n-x^2-y^2 ;
            if issqr(z2) then
                z := sqrt(z2) ;
                if z >= y then
                    a := a+1 ;
                end if;
            end if;
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Feb 12 2017

Length[PowersRepresentations[ #, 3, 2]] & /@ Range[0, 104]
e[0,r_,s_,m_]=0;e[n_,r_,s_,m_]:=Length[Select[Divisors[n],Mod[ #,m]==r &]]-Length[Select[Divisors[n],Mod[ #,m]==s &]];alpha[n_]:=5delta[n]+3delta[1/2 n]+4delta[1/3n];beta[n_]:=4e[n,1,3,4]+3e[n,1,7,8]+3e[n,3,5,8];delta[n_]:=If[IntegerQ[Sqrt[n]],1,0];f[n_]:=Table[n-k^2, {k,1,Floor[Sqrt[n]]}]; gamma[n_]:=2 Plus@@(e[ #,1,3,4] &/@f[n]);p3[n_]:=1/12(alpha[n]+beta[n]+gamma[n]);p3[ # ] &/@Range[0,104]
(* Ant King, Oct 15 2010 *)
a[ n_] := If[ n < 0, 0, Sum[ Boole[ n == i^2 + j^2 + k^2], {i, 0, Sqrt[n]}, {j, 0, i}, {k, 0, j}]]; (* Michael Somos, Aug 15 2015 *)

{a(n) = if( n<0, 0, sum( i=0, sqrtint(n), sum( j=0, i, sum( k=0, j, n == i^2 + j^2 + k^2))))}; /* Michael Somos, Jun 05 2012 */

import collections; a = collections.Counter(i*i + j*j + k*k for i in range(100) for j in range(i+1) for k in range(j+1)) # David Radcliffe, Apr 15 2019

Let e(n,r,s,m) be the excess of the number of n's r(mod m) divisors over the number of its s (mod m) divisors, and let delta(n)=1 if n is a perfect square and 0 otherwise. Then, if we define alpha(n) = 5*delta(n) + 3*delta(n/2) + 4*delta(n/3), beta(n) = 4*e(n,1,3,4) + 3*e(n,1,7,8) + 3*e(n,3,5,8), gamma(n) = 2*Sum_{1<=k^2Ant King, Oct 15 2010

Name clarified by Wolfdieter Lang, Apr 08 2013

A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 2, 3, 3, 3, 2, 1, 1, 0, 1, 1, 2, 3, 3, 4, 4, 2, 1, 1, 0, 1, 1, 2, 3, 4, 5, 5, 4, 1, 1, 1, 0, 1, 1, 2, 4, 5, 7, 9, 6, 2, 4, 2, 1, 0

Offset: 0

Alois P. Heinz, Feb 17 2015

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 2,  2,  2,  2,  2,  2,  2, ...
  0, 1, 1, 2, 2,  2,  3,  3,  3,  4,  4, ...
  0, 1, 1, 1, 2,  3,  3,  4,  5,  5,  6, ...
  0, 1, 2, 2, 3,  4,  5,  7,  8,  9, 11, ...
  0, 1, 1, 2, 4,  5,  9, 10, 11, 15, 17, ...
  0, 1, 1, 2, 4,  6,  9, 13, 18, 21, 27, ...
  0, 1, 1, 1, 2,  7,  9, 16, 25, 30, 41, ...
  0, 1, 1, 4, 6,  8, 18, 27, 36, 52, 68, ...
  0, 1, 2, 2, 7, 13, 23, 36, 51, 70, 94, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000012, A063014, A016727, A065458, A065459, A065460, A065461, A065462, A255213, A255214.
Main diagonal gives A105152.
Cf. A302996.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1 and n<=t, 1,
      (j-> `if`(t*jn, 0, b(n-j, i, t-1))))(i^2))
    end:
A:= (n, k)-> b(n^2, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1 && n <= t, 1, Function[j, If[t*jn, 0, b[n-j, i, t-1]]]][i^2]]; A[n_, k_] := b[n^2, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

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

Original entry on oeis.org

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]

A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 2, 6, 7, 6, 4, 8, 10, 14, 2, 11, 14, 13, 7, 23, 15, 17, 4, 24, 21, 31, 10, 25, 37, 28, 2, 46, 29, 49, 14, 38, 35, 61, 7, 45, 62, 49, 15, 93, 47, 57, 4, 72, 67, 97, 21, 71, 84, 101, 10, 119, 70, 86, 37, 92, 79, 165, 2, 138, 127, 109, 29, 168, 140, 121, 14

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=3 because 25 produces {0,0,0,5}, {0,0,3,4}, {1,2,2,4}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A063014, A016727, A065459.

Column k=4 of A255212.

N:= 100:
R:= Vector(N,1):
for a from 0 to N do
  for b from a to floor(sqrt(N^2-a^2)) do
     for c from b to floor(sqrt(N^2-a^2-b^2)) do
       q:= a^2 + b^2 + c^2;
       for f in numtheory:-divisors(q) do
          if f^2 + 2*f*c <= q and (f + q/f)::even then
             r:= (f + q/f)/2;
             if r <= N then R[r]:= R[r]+1 fi;
          fi
od od od od:
convert(R,list); # Robert Israel, Feb 16 2015

Length/@Table[SumOfSquaresRepresentations[4, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065459 Number of inequivalent (ordered) solutions to n^2 = sum of 5 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 13, 12, 13, 17, 25, 22, 27, 31, 35, 38, 46, 49, 61, 61, 61, 73, 92, 83, 112, 106, 118, 127, 147, 138, 185, 175, 178, 198, 239, 212, 254, 262, 298, 294, 341, 304, 404, 376, 385, 432, 483, 441, 539, 517, 560, 551, 680, 587, 745, 693, 698

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=4 because 25 produces {0,0,0,0,5}, {0,0,0,3,4}, {0,1,2,2,4}, {2,2,2,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..750

Cf. A063014, A016727.

Column k=5 of A255212.

Length/@Table[SumOfSquaresRepresentations[5, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

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

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]

A065461 Number of inequivalent (ordered) solutions to n^2 = sum of 7 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 13, 16, 27, 36, 43, 58, 72, 99, 130, 146, 178, 254, 265, 342, 417, 507, 540, 726, 745, 975, 1092, 1289, 1338, 1845, 1751, 2246, 2447, 2948, 2852, 3932, 3638, 4728, 4868, 5778, 5618, 7659, 6887, 8891, 8887, 10825, 10109, 13712

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(4)=4 because 16 produces {0,0,0,0,0,0,4}, {0,0,0,2,2,2,2}, {0,0,1,1,1,2,3}, {1,1,1,1,2,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A063014, A016727.

Column k=7 of A255212.

Table[Length[PowersRepresentations[n^2,7,2]],{n,0,50}] (* Harvey P. Dale, Sep 08 2020 *)

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

Previous Mathematica program replaced by Harvey P. Dale, Sep 08 2020

A065460 Number of inequivalent (ordered) solutions to n^2 = sum of 6 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 9, 9, 9, 18, 23, 24, 29, 37, 53, 62, 59, 77, 116, 106, 130, 156, 199, 192, 221, 257, 342, 336, 384, 402, 577, 501, 599, 639, 835, 774, 910, 912, 1220, 1113, 1378, 1298, 1703, 1530, 1907, 1862, 2398, 2094, 2471, 2393, 3356, 2765, 3543

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=5 because 25 produces {0,0,0,0,0,5}, {0,0,0,0,3,4}, {0,0,1,2,2,4}, {0,2,2,2,2,3}, {1,1,1,2,3,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A063014, A016727.

Column k=6 of A255212.

Length/@Table[SumOfSquaresRepresentations[6, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065462 Number of inequivalent (ordered) solutions to n^2 = sum of 8 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 18, 25, 36, 51, 73, 90, 133, 169, 223, 295, 380, 452, 603, 763, 903, 1115, 1385, 1668, 2025, 2398, 2811, 3535, 4011, 4683, 5503, 6724, 7316, 8684, 9946, 11844, 12994, 15091, 16712, 20493, 21663, 24975, 27536, 33079, 34654, 39957, 43315

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(4)=5 because 16 produces {0,0,0,0,0,0,0,4}, {0,0,0,0,2,2,2,2}, {0,0,0,1,1,1,2,3}, {0,1,1,1,1,2,2,2}, {1,1,1,1,1,1,1,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A016727, A063014.

Column k=8 of A255212.

Length/@Table[SumOfSquaresRepresentations[8, (k)^2], {k, 36}]

a(0), a(37)-a(47) from Alois P. Heinz, Feb 16 2015

cp's OEIS Frontend

A000164 Number of partitions of n into 3 squares (allowing part zero).

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A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

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A065459 Number of inequivalent (ordered) solutions to n^2 = sum of 5 squares of integers >= 0.

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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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Maple

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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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Mathematica

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PARI

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