cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A322039 Expansion of (1 + x)^2 / ((1 - x)^2*(1 + 2*x)^2).

Original entry on oeis.org

1, 0, 4, -4, 16, -28, 72, -148, 336, -716, 1560, -3332, 7136, -15164, 32168, -67956, 143216, -300972, 631096, -1320420, 2757376, -5747740, 11961544, -24855124, 51574416, -106877068, 221210712, -457334468, 944495136, -1948642556
Offset: 0

Views

Author

N. J. A. Sloane, Dec 03 2018

Keywords

Comments

Connected with tiling of torus by squares (see A322038).

Links

Crossrefs

Cf. A322038, A322040.

Programs

  • Mathematica
    LinearRecurrence[{-2, 3, 4, -4}, {1, 0, 4, -4}, 100] (* Amiram Eldar, Dec 04 2018 *)
  • PARI
    Vec((1 + x)^2 / ((1 - x)^2*(1 + 2*x)^2) + O(x^40)) \\ Colin Barker, Dec 04 2018

Formula

From Colin Barker, Dec 04 2018: (Start)
a(n) = -2*a(n-1) + 3*a(n-2) + 4*a(n-3) - 4*a(n-4) for n>3.
a(n) = (16 + 11*(-2)^n + 3*(4+(-2)^n)*n) / 27.
(End)

A322040 Expansion of (1 + x)^2 / ((1 - x)^2*(1 + 2*x + 2*x^2)^2).

Original entry on oeis.org

1, 0, 0, 4, -4, 4, 8, -20, 32, -12, -40, 124, -160, 68, 232, -628, 816, -300, -1160, 3100, -3904, 1380, 5640, -14676, 18256, -6156, -26472, 67900, -83488, 27268, 121640, -308276, 375920, -119532, -549448, 1379932, -1671424, 520100, 2449480
Offset: 0

Views

Author

N. J. A. Sloane, Dec 03 2018

Keywords

Comments

Connected with tiling of torus by squares (see A322038).

Links

Crossrefs

Cf. A322038, A322039.

Programs

  • Mathematica
    LinearRecurrence[{-2, -1, 4, 4, 0, -4}, {1, 0, 0, 4, -4, 4}, 100] (* Amiram Eldar, Dec 04 2018 *)
CoefficientList[Series[(1+x)^2/((1-x)^2(1+2x+2x^2)^2),{x,0,40}],x] (* Harvey P. Dale, Jan 20 2021 *)
  • PARI
    Vec((1 + x)^2 / ((1 - x)^2*(1 + 2*x + 2*x^2)^2) + O(x^40)) \\ Colin Barker, Dec 04 2018

Formula

a(n) = -2*a(n-1) - a(n-2) + 4*a(n-3) + 4*a(n-4) - 4*a(n-6) for n>5. - Colin Barker, Dec 04 2018

A322041 Triangle read by rows: let E denote the standard triangular 6-valent grid in the plane, regarded as a graph with the Eisenstein integers as vertices; row n gives the coordination sequence of the quotient graph E/nE.

Original entry on oeis.org

1, 1, 3, 1, 6, 2, 1, 6, 9, 0, 1, 6, 12, 6, 0, 1, 6, 12, 15, 2, 0, 1, 6, 12, 18, 12, 0, 0, 1, 6, 12, 18, 21, 6, 0, 0, 1, 6, 12, 18, 24, 18, 2, 0, 0, 1, 6, 12, 18, 24, 27, 12, 0, 0, 0, 1, 6, 12, 18, 24, 30, 24, 6, 0, 0, 0, 1, 6, 12, 18, 24, 30, 33, 18, 2, 0, 0, 0, 1, 6, 12, 18, 24, 30, 36, 30, 12, 0, 0, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane, Dec 05 2018; corrected and extended Dec 06 2018

Keywords

Comments

The Eisenstein integers E are the complex numbers r+s*omega, where r, s in Z and omega = exp(2*Pi*i/3) is a complex cube root of unity.
Denote the entries in the triangle by T(n,k), for n >= 1, 0 <= k <= n-1. Then T(n,k) <= 6*k for k >= 1, and Sum_{k=0..n-1} T(n,k) = n^2.
When E is regarded as a lattice in R^2, E/nE has packing radius roughly n/2, but covering radius roughly n/sqrt(3) > n/2 (see Conway-Sloane, Chapter 4). This means that as n increases, the number of terms in the n-th row of the triangle will increase linearly with n. The largest k such that T(n,k) is nonzero is A322042(n), which is conjecturally n - ceiling(n/3).

Examples

			The first 18 rows are
1 [1]
2 [1, 3]
3 [1, 6, 2]
4 [1, 6, 9, 0]
5 [1, 6, 12, 6, 0]
6 [1, 6, 12, 15, 2, 0]
7 [1, 6, 12, 18, 12, 0, 0]
8 [1, 6, 12, 18, 21, 6, 0, 0]
9 [1, 6, 12, 18, 24, 18, 2, 0, 0]
10 [1, 6, 12, 18, 24, 27, 12, 0, 0, 0]
11 [1, 6, 12, 18, 24, 30, 24, 6, 0, 0, 0]
12 [1, 6, 12, 18, 24, 30, 33, 18, 2, 0, 0, 0]
13 [1, 6, 12, 18, 24, 30, 36, 30, 12, 0, 0, 0, 0]
14 [1, 6, 12, 18, 24, 30, 36, 39, 24, 6, 0, 0, 0, 0]
15 [1, 6, 12, 18, 24, 30, 36, 42, 36, 18, 2, 0, 0, 0, 0]
16 [1, 6, 12, 18, 24, 30, 36, 42, 45, 30, 12, 0, 0, 0, 0, 0]
17 [1, 6, 12, 18, 24, 30, 36, 42, 48, 42, 24, 6, 0, 0, 0, 0, 0]
18 [1, 6, 12, 18, 24, 30, 36, 42, 48, 51, 36, 18, 2, 0, 0, 0, 0, 0]
...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. Fig. 7.1, p. 199. Illustrates row 2 (note that E/2E is isomorphic to GF(4)).

Links

Crossrefs

The rows converge to A008458.
Cf. A322038 (an analog for the square grid), A322042.

Programs

  • Maple
    # We work in a fundamental region for E/nE and calculate the edge-distance of each point to the nearest point of nE.
hist:=proc(n) local A,i,j,m,d1,d2,d3,d4;
A:=Array(0..n,0);
for i from 0 to n-1 do
for j from 0 to n-1 do
d1:=i+j; d2:=n-i; d3:=2*n-i-j; d4:=n-j;
if i+j

Formula

Examination of the first 36 rows (see link) shows an obvious quasi-periodic structure. Call an entry T(n,k) "full" if k=0 or T(n,k)=6*k. Then it appears that column k>0 is full starting at n=2k+1. It also appears that the number of trailing 0's is floor((n-1)/3) (see A322042). Combining these two observations suggests that the rows of the triangle are quasi-periodic with period 6.
One can now formulate a specific conjecture for what row n is, for each of the six residue classes of n mod 6.
For example, suppose n=6t. Then it appears that row n is [1, 6, 18, 24, ..., 18t-6, 18t-3, 18(t-1), 18(t-2), 18(t-3), ..., 36, 18, 2, 0 (2t-1 times)].
For t=3, for example, we get:
[1, 6, 12, 18, 24, 30, 36, 42, 48, 51, 36, 18, 2, 0, 0, 0, 0, 0]
There are similar conjectures for n = 6t+1, ..., 6t+5.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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