A267326 -id:A267326 - cp's OEIS frontend

A264390 Partial sums of A267326.

8, 32, 136, 160, 408, 720, 1176, 1200, 2168, 2912, 3976, 4288, 5752, 7120, 10344, 10368, 12824, 15728, 18776, 19520, 25448, 28640, 33064, 33376, 39624, 44016, 52760, 54128, 61096, 70768, 78712, 78736, 92568, 99936, 114072, 116976, 128232, 137376, 156408

Offset: 1

Christopher Heiling, Jan 12 2016

			For n = 2 the a(n) = 32 integral solutions of x^2 + y^2 + z^2 + t^2 <= 2^2 are: {x,y,z,t} = {{0,0,0,1}; {0,0,1,0}; {0,1,0,0}; {1,0,0,0}; {0,0,0,-1}; {0,0,-1,0}; {0,-1,0,0}; {-1,0,0,0}; {0,0,0,2}; {0,0,0,-2}; {0,0,2,0}; {0,0,-2,0}; {0,2,0,0}; {0,-2,0,0}; {2,0,0,0}; {-2,0,0,0}; {1,1,1,1}; {1,1,1,-1}; {1,1,-1,1}; {1,-1,1,1}; {-1,1,1,1}; {1,1,-1,-1}; {1,-1,1,-1}; {-1,1,1,-1}; {1,-1,-1,1}; {-1,1,-1,1}; {1,-1,-1,-1}; {-1,1,-1,-1}; {-1,-1,1,-1}; {-1,-1,1,-1}; {-1,-1,-1,1}; {-1,-1,-1,-1}}.

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

Partial sums of A267326.

Cf. A000118, A046897.

#A264390
terms := 42:
(add(q^(m^2), m = -terms..terms))^4:
seq(add(coeff(%, q, k^2), k = 1..n), n = 1..terms); # Peter Bala, Jan 15 2016

a000118(k) = if(k<1, k==0, 8 * sumdiv( k, d, if( d%4, d)));
a(n) = sum(k=1, n, a000118(k^2)); \\ Altug Alkan, Jan 19 2016

a(n) = Sum_{k = 1..n} A000118(k^2).

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.

A319617 Number of Integer solutions to w^2 + x^2 + y^2 + z^2 < n^2; number of lattice points inside a 4-sphere of radius n.

Original entry on oeis.org

0, 1, 65, 321, 1257, 2873, 6265, 11377, 20161, 31665, 48945, 71401, 102041, 139481, 188753, 247329, 323697, 409457, 516121, 640393, 789161, 955793, 1153025, 1376305, 1637929, 1921049, 2252889, 2615673, 3033665, 3483633, 3990753, 4547945, 5173145, 5840393, 6589945, 7395921, 8287297, 9238001, 10281977, 11402457, 12633145, 13929377

Offset: 0

Brian J. Harrild, Sep 24 2018

			For n=2 there are 65 lattice points in Z^4 such that w^2+x^2+y^2+x^2 < 4

a(n) = A055410(n) - A267326(n).

for n in range (0,51):
    NumPoints=0
    for w in range (-n,n+1):
        for x in range (-n,n+1):
            for y in range (-n,n+1):
                for z in range (-n,n+1):
                    if w**2+x**2+y**2+z**2

cp's OEIS Frontend

A264390 Partial sums of A267326.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A319617 Number of Integer solutions to w^2 + x^2 + y^2 + z^2 < n^2; number of lattice points inside a 4-sphere of radius n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python