Christopher Heiling - cp's OEIS frontend

User: Christopher Heiling

Christopher Heiling has authored 3 sequences.

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).

A264390 Partial sums of A267326.

Original entry on oeis.org

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).

A267326 Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2).

Original entry on oeis.org

1, 8, 24, 104, 24, 248, 312, 456, 24, 968, 744, 1064, 312, 1464, 1368, 3224, 24, 2456, 2904, 3048, 744, 5928, 3192, 4424, 312, 6248, 4392, 8744, 1368, 6968, 9672, 7944, 24, 13832, 7368, 14136, 2904, 11256, 9144, 19032, 744, 13784, 17784, 15144, 3192

Offset: 0

Christopher Heiling, Jan 13 2016

For all pair of relatively prime numbers k, m this sequence is multiplicative with a factor of 8: a(k*m) = 8*a(k)*a(m). - Christopher Heiling, Apr 02 2017

			For n = 2 the a(n) = 24 solutions of x^2 + y^2 + z^2 + t^2 = 2^2 are:
{x,y,z,t} = {{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}}.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..150 from Christopher Heiling)

Cf. A000118.
Partial sums of this sequence give A264390.
Column k=4 of A302996.

terms := 42:
(add(q^(m^2), m = -terms..terms))^4:
seq(coeff(%, q, n^2), n = 0..terms); # Peter Bala, Jan 15 2016

a[n_] := SquaresR[4, n^2];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 18 2023 *)

a(n) = A264390(n) - A264390(n-1) for n > 1 and a(1) = A264390(1) = 2*D.
a(n) = 8*sigma(n^2) if n is odd else 24*sigma(m(n^2)), where sigma(n) = A000203(n) and m(n) = A000265(n) is the largest odd divisor of n. - Peter Bala, Jan 15 2016
a(p^(k+1)) = 8*(p^2 *a(p^k)+p+1) for p prime. In particular a(p) = 8*(p^2+p+1). - Christopher Heiling, Apr 02 2017

a(0)=1 prepended by Alois P. Heinz, Mar 10 2023

cp's OEIS Frontend

User: Christopher Heiling

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A264390 Partial sums of A267326.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A267326 Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions