A016725 -id:A016725 - cp's OEIS frontend

A267309 Number of discrete vectors with integral components and integral length <= n in a 3-dimensional vectorspace (Partial sums of A016725).

6, 12, 42, 48, 78, 108, 162, 168, 270, 300, 378, 408, 486, 540, 690, 696, 798, 900, 1026, 1056, 1326, 1404, 1554, 1584, 1734, 1812, 2130, 2184, 2358, 2508, 2706, 2712, 3102, 3204, 3474, 3576, 3798, 3924, 4314, 4344, 4590, 4860, 5130, 5208, 5718

Offset: 1

Christopher Heiling, Jan 19 2016

nonn

This sequence is Z_3(n), where Z_D(n) counts all vectors with integral components and length in a D-dimensional vectorpace within a certain radius. This sequence represents partial sums of A016725.

			For n = 2 the a(n)= 12 integral solutions of x^2 + y^2 + z^2 <= 2^2 are: {x,y,z} = {{0,0,1}; {0,1,0}; {1,0,0}; {0,0,-1}; {0,-1,0}; {-1,0,0}; {0,0,2}; {0,2,0}; {2,0,0}; {0,0,-2}; {0,-2,0}; {-2,0,0}}.

Christopher Heiling, Table of n, a(n) for n = 1..150

Cf. A005875, A016725.

a(n) = Sum_{k=1..n} A005875(k^2).

a(n) = Sum_{k=1..n} A016725(k).

A267651 Duplicate of A016725.

Original entry on oeis.org

6, 6, 30, 6, 30, 30, 54, 6, 102, 30, 78, 30, 78, 54, 150, 6, 102, 102, 126, 30, 270, 78, 150, 30, 150, 78, 318, 54, 174, 150, 198, 6, 390, 102, 270, 102, 222, 126, 390, 30, 246, 270, 270, 78, 510, 150, 294, 30, 390, 150, 510, 78, 318, 318, 390, 54, 630

Offset: 1

dead

A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 4, 2, 0, 1, 8, 6, 4, 2, 0, 1, 10, 24, 30, 4, 2, 0, 1, 12, 90, 104, 6, 12, 2, 0, 1, 14, 252, 250, 24, 30, 4, 2, 0, 1, 16, 574, 876, 730, 248, 30, 4, 2, 0, 1, 18, 1136, 3542, 4092, 1210, 312, 54, 4, 2, 0, 1, 20, 2034, 12112, 18494, 7812, 2250, 456, 6, 4, 2, 0

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  0,  2,   4,   6,    8,    10,  ...
  0,  2,   4,   6,   24,    90,  ...
  0,  2,   4,  30,  104,   250,  ...
  0,  2,   4,   6,   24,   730,  ...
  0,  2,  12,  30,  248,  1210,  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.
Main diagonal gives A232173.
Cf. A000122, A122141, A255212, A286815.

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
A:= (n, k)-> b(n^2, k):
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 10 2023

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.

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]

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]

A386315 Number of points in a face-centered cubic lattice that intersect a sphere of radius n centered on a point in the lattice.

Original entry on oeis.org

1, 12, 12, 36, 12, 84, 36, 108, 12, 108, 84, 132, 36, 180, 108, 252, 12, 204, 108, 228, 84, 324, 132, 300, 36, 444, 180, 324, 108, 372, 252, 396, 12, 396, 204, 756, 108, 468, 228, 540, 84, 492, 324, 516, 132, 756, 300, 588, 36, 780, 444, 612, 180, 660, 324

Offset: 0

Charles L. Hohn, Aug 15 2025

For all n > 0, the points at the 4 90-degree rotations of [n, 0, 0] and the eight 90-degree rotations and vertical reflections of [n/2, n/2, n*sqrt(1/2)] form the 12 vertices of a cuboctahedron (shown in red in the Links example). Points that otherwise lie on an axis plane have 24-point symmetry (green), and all other points have 48-point symmetry (blue). Thus, a(n) for all n > 0 are odd multiples of 12.
The number of noncongruent points for each sphere radius n (points within or on vertices or edges of each symmetry region in the Links example) gives A387222, the number of those points that are primitive for radius n (darker colors) gives A387223 for odd sphere radii, and the total primitive count divided by 12 gives A278081 for odd sphere radii. Examples of nonprimitive points include [3, 0, 0] and [3/2, 3/2, 3*sqrt(1/2)] for a(3), which reduce to a(1) primitive points [1, 0, 0] and [1/2, 1/2, sqrt(1/2)] respectively.
Analog for the simple cubic lattice is A016725.

			a(3) = 36, which is the sum of 4 90-degree rotations of [3, 0, 0], 8 90-degree rotations and vertical reflections of [3/2, 3/2, 3*sqrt(1/2)] and [1, 0, 4*sqrt(1/2)], and 16 90-degree rotations and vertical and horizontal reflections of [5/2, 3/2, sqrt(1/2)].

Charles L. Hohn, a(0) to a(15), animated - see Comments

Cf. A004015, A016725.

Cf. A387222, A387223, A278081.

a(n)={my(c=0); for(x=0, n, for(y=0, min(x, sqrtint(n^2-x^2)), for(o=0, 1, my(m=2*(n^2-(x+o/2)^2-(y+o/2)^2)); if(!issquare(m), next); my(z=sqrtint(m)); if(z>=0 && z%2==o, c+=2^(4-if(!z, 1)-if(x==y, 1)-if(!min(x, y) && !o, 1)-if(!vecmax([x, y, z, o]), 1)))))); c}

a(n)={if(!n, return(1)); my(f=Vec(factor(n)), o=12, r=o); for(i=if(#f[1] && f[1][1]==2, 2, 1), #f[1], my(m=if(f[1][i]%8>=4, 2)); f[2][i]++; while(f[2][i]--, o=o*f[1][i]+r*m); r=o); o}

a(n) = A004015(n^2).
a(2*n) = a(n).
a(p*n) = p*a(n) where p is a prime and p mod 8 is in {1, 3}.
a(p*n) = p*a(n) + 2*a(n/p^c) where p is a prime, p mod 8 is in {5, 7}, and c is the count of prime factors p in n.

A078183 Number of solutions to x^2 + y^2 + z^2 < n^2; number of lattice points inside a sphere of radius n.

Original entry on oeis.org

0, 1, 27, 93, 251, 485, 895, 1365, 2103, 2969, 4139, 5497, 7123, 9093, 11459, 13997, 17071, 20377, 24303, 28545, 33371, 38641, 44395, 50733, 57747, 65117, 73447, 82201, 91911, 101769, 112931, 124289, 137059, 150165, 164415, 179309, 195167

Offset: 0

T. D. Noe, Nov 21 2002

nonn

Cf. A000605, A016725.

s = 0; Table[s = s + Sum[SquaresR[3, k], {k, (n - 1)^2, n^2 - 1}], {n, 0, 50}]

a(n) = A000605(n) - A016725(n)

cp's OEIS Frontend

A267309 Number of discrete vectors with integral components and integral length <= n in a 3-dimensional vectorspace (Partial sums of A016725).

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A267651 Duplicate of A016725.

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A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

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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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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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PARI

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A386315 Number of points in a face-centered cubic lattice that intersect a sphere of radius n centered on a point in the lattice.

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PARI

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A078183 Number of solutions to x^2 + y^2 + z^2 < n^2; number of lattice points inside a sphere of radius n.

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