A063691 -id:A063691 - cp's OEIS frontend

A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

1, 0, 0, 10, 0, 0, 45, 0, 10, 120, 0, 90, 210, 0, 360, 262, 45, 840, 300, 360, 1260, 480, 1260, 1350, 1015, 2520, 1560, 2200, 3150, 2880, 4186, 2880, 5430, 6240, 3780, 8300, 7080, 7920, 11160, 7320, 13257, 14640, 10600, 16470, 18570, 18240, 19620, 22230, 25135, 27720, 28020, 28480, 38160

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

David A. Corneth, Table of n, a(n) for n = 10..10009
Index entries for sequences related to sums of squares

Cf. A000144, A000290, A010052, A025434, A045852, A063691, A063725, A063730, A340481, A340905, A340906, A340915, A340946.

Column k=10 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^10/1024, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (theta_3(x) - 1)^10 / 1024, where theta_3() is the Jacobi theta function.

A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

Original entry on oeis.org

1, 6, 24, 77, 216, 552, 1315, 2964, 6387, 13255, 26640, 52074, 99336, 185430, 339483, 610709, 1081227, 1886484, 3247502, 5521365, 9279624, 15429149, 25397088, 41412030, 66928700, 107265576, 170556654, 269164346, 421765920, 656419080, 1015044526, 1559950185, 2383284894

Offset: 3

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A002448, A004404, A014968, A015128, A063691, A319552, A327381, A338223, A341221, A341365, A341366, A341367, A341368, A341369, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^3/8, {x, 0, nmax}], x] // Drop[#, 3] &
nmax = 35; CoefficientList[Series[(1/8) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

G.f.: (1/8) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^3.

a(n) ~ A319552(n)/8 ~ 3*exp(Pi*sqrt(3*n)) / (512*n^(3/2)). - Vaclav Kotesovec, Feb 20 2021

A211639 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2+x^2+y^2<=n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 4, 7, 7, 10, 11, 11, 17, 17, 17, 20, 23, 26, 26, 32, 35, 35, 38, 38, 44, 48, 48, 54, 60, 60, 60, 66, 69, 75, 78, 78, 87, 87, 87, 96, 102, 105, 108, 114, 120, 120, 121, 127, 133, 139, 139, 145, 157, 157, 163, 169, 169, 178, 178, 184, 196

Offset: 0

Clark Kimberling, Apr 18 2012

nonn

For a guide to related sequences, see A211422.

Cf. A211638, A211422. Partial sums of A063691.

t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w^2 + x^2 + y^2 <= n, s = s + 1],
        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 80]]     (* A211639 *)
(* Peter J. C. Moses, Apr 13 2012 *)

a(n) = A211638(n)+A063691(n). - R. J. Mathar, Jan 07 2015

G.f.: (theta_3(x) - 1)^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018

A237707 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

1, 4, 7, 10, 11, 17, 20, 23, 26, 32, 35, 38, 44, 48, 54, 60, 66, 69, 75, 78, 87, 96, 102, 105, 108, 114, 120, 121, 127, 133, 139, 145, 157, 163, 169, 178, 184, 196, 202, 214, 217, 220, 232, 238, 241, 244, 256, 263, 266, 278, 284, 296, 299, 308, 314, 329, 332

Offset: 1

Rajan Murthy, Feb 11 2014

nonn

			When the radius of the sphere reaches 3^(1/2), one cube is completely within the sphere. When the radius reaches 6^(1/2), four cubes are completely within the sphere.

Rajan Murthy, Table of n, a(n) for n = 1..200
Rajan Murthy, Table of n, a(n), and squared radius for n = 1..200
Rajan Murthy, Scilab program for this sequence
Charles R Greathouse IV, Illustration of this sequence

The radii corresponding to the terms are given by the square roots of A000408 starting with squared radius 3.
Cf. A232499 (2-dimensional analog).
Partial sums of A014465 and A063691 (but then with repeated terms omitted).

(* Illustrates the sequence *)
Cube[x_,y_,z_]:=Cuboid[{x-1,y-1,z-1},{x,y,z}]
Cubes[r_]:=Cube@@#&/@Select[Flatten[Table[{x,y,z},{x,1,r},{y,1,r},{z,1,r}],2],Norm[#]<=r&]
Draw[r_]:=Graphics3D[Union[Cubes[r],{{Green, Opacity[0.3], Sphere[{0,0,0},r]}}],PlotRange->{{0,r},{0,r},{0,r}},ViewPoint->{r,3r/4,3r/5}];
Draw/@Sqrt/@{3,6,9,11,12,14} (* Charles R Greathouse IV, Mar 12 2014 *)

Scilab
```
// See Murthy link.
```

a(n) ~ (Pi*sqrt(30)/25)*n^(3/2). - Charles R Greathouse IV, Mar 14 2014

Duplicate terms deleted by Rajan Murthy, Mar 06 2014

Terms a(36) and beyond added from b-file by Andrew Howroyd, Feb 27 2018

A321429 Expansion of Product_{i>0, j>0, k>0} (1 + x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 3, 0, 0, 6, 0, 3, 7, 0, 9, 13, 0, 18, 19, 3, 39, 28, 9, 66, 42, 33, 105, 68, 78, 168, 111, 153, 261, 185, 285, 411, 325, 483, 636, 563, 798, 1017, 949, 1275, 1620, 1556, 2061, 2547, 2500, 3303, 4008, 3969, 5226, 6216, 6252, 8301, 9534, 9784, 12984

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A063691, A321381, A321428.

G.f.: Product_{k>0} (1 + x^k)^A063691(k).

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

A321432 Expansion of Product_{i>0, j>0, k>0} (1 - x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, -1, 0, 0, -3, 0, 0, 0, 0, -3, 5, 0, -3, 7, 0, 12, -7, -3, 21, -14, 3, -6, 6, 27, -57, 22, 6, -36, 15, -75, 87, -17, -111, 99, -71, 75, -90, -91, 324, -225, 23, 57, -36, 332, -543, 333, 374, -417, 342, -473, 720, 18, -1132, 1227, -330, 202, -414, -846, 2357, -1998

Offset: 0

Seiichi Manyama, Nov 09 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A063691, A321429, A321433.

G.f.: Product_{k>0} (1 - x^k)^A063691(k).

A321433 Expansion of Product_{i>0, j>0, k>0} 1/(1 - x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 4, 0, 0, 7, 0, 3, 14, 0, 9, 23, 0, 21, 45, 3, 48, 72, 12, 96, 124, 39, 180, 204, 105, 327, 343, 225, 585, 572, 468, 1011, 976, 903, 1719, 1662, 1689, 2895, 2844, 3018, 4836, 4791, 5355, 8013, 8061, 9234, 13182, 13429, 15714, 21573, 22257, 26346

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A321432.

Cf. A063691, A321429, A321431.

G.f.: Product_{k>0} 1/(1 - x^k)^A063691(k).

A211638 Number of ordered triples (w, x, y) with all terms in {1, ..., n} and w^2 + x^2 + y^2 < n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 4, 4, 4, 7, 7, 10, 11, 11, 17, 17, 17, 20, 23, 26, 26, 32, 35, 35, 38, 38, 44, 48, 48, 54, 60, 60, 60, 66, 69, 75, 78, 78, 87, 87, 87, 96, 102, 105, 108, 114, 120, 120, 121, 127, 133, 139, 139, 145, 157, 157, 163, 169, 169, 178, 178, 184

Offset: 0

Clark Kimberling, Apr 18 2012

nonn

For a guide to related sequences, see A211422.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A211422.

t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w^2 + x^2 + y^2 < n, s = s + 1],
        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 80]]     (* A211638 *)
(* Peter J. C. Moses, Apr 13 2012 *)

first(n) = {n = max(n, 2); n-=2; my(res = vector(n), v = vector(n)); forvec(x = vector(3, i, [1,sqrtint(n)]), c = sum(i = 1, 3, x[i]^2); if(c <= n, v[c]++)); for(i = 2, #v, v[i]+=v[i-1]); concat([0,0],v)} \\ David A. Corneth, Jun 16 2023

a(n) + A063691(n) = A211639(n). - R. J. Mathar, Jun 16 2023

a(n) = A211639(n-1). - R. J. Mathar, Jun 16 2023

A347710 Number of compositions (ordered partitions) of n into at most 3 squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 0, 1, 4, 2, 3, 1, 2, 6, 0, 1, 5, 4, 3, 2, 6, 3, 0, 3, 3, 8, 4, 0, 8, 6, 0, 1, 6, 5, 6, 4, 2, 9, 0, 2, 11, 6, 3, 3, 8, 6, 0, 1, 7, 9, 6, 2, 8, 12, 0, 6, 6, 2, 9, 0, 8, 12, 0, 1, 10, 12, 3, 5, 12, 6, 0, 4, 5, 14, 7, 3, 12, 6, 0, 2

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A002654, A006456, A063691, A337165, A347711, A347712, A347713.

a(n) = Sum_{k=0..3} A337165(n,k). - Alois P. Heinz, Sep 10 2021

cp's OEIS Frontend

A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

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A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

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A211639 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2+x^2+y^2<=n.

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A237707 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

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A321429 Expansion of Product_{i>0, j>0, k>0} (1 + x^(i^2 + j^2 + k^2)).

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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

Mathematica

Formula

A321432 Expansion of Product_{i>0, j>0, k>0} (1 - x^(i^2 + j^2 + k^2)).

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A321433 Expansion of Product_{i>0, j>0, k>0} 1/(1 - x^(i^2 + j^2 + k^2)).

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A211638 Number of ordered triples (w, x, y) with all terms in {1, ..., n} and w^2 + x^2 + y^2 < n.

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