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.

A244500 Number T(n, k) of ways to place k points on an n X n X n triangular grid so that no pair of them has distance sqrt(3). Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 6, 12, 8, 1, 10, 36, 55, 33, 9, 1, 15, 87, 248, 378, 339, 187, 63, 12, 1, 1, 21, 180, 820, 2190, 3606, 3716, 2340, 825, 125, 1, 28, 333, 2212, 9110, 24474, 43928, 53018, 42774, 22792, 7945, 1764, 196, 1, 36, 567, 5163, 30300, 121077, 339621
Offset: 1

Views

Author

Heinrich Ludwig, Jun 29 2014

Keywords

Comments

In the following triangular grid points x have Euclidean distance sqrt(3) from point o. It is the second closest distance possible among grid points.
x
. .
. o .
x . . x
Triangle T(n, k) is irregular: 0 <= k <= max(n), where max(n), the maximal number of points that can be placed on the grid, is:
for n = 3j-2: max(n) = A000326(j) = j(3j-1)/2;
for n = 3j-1 or n = 3j: max(n) = A045943(j) = 3j(j+1)/2; j = 1,2,3,...
Empirical: (1) The number of ways to place the maximal number of points for grid sizes n = 3j are cubes of Catalan numbers, i.e., for n = 3j: T(n, max(n)) = C(j+1)^3 = A033536(j+1). (2) For n = 3j-2: T(n, max(n)) = A244506(n) = A244507^2(n). (3) For n = 3j-1: T(n, max(n)) = A000012(n) = 1 and T(n, max(n)-1) = 3j^2.
Row n is also the coefficients of the independence polynomial of the n-triangular honeycomb acute knight graph. - Eric W. Weisstein, May 21 2017

Examples

			On an 8 X 8 X 8 grid there is T(8, 18) = 1 way to place 18 points (x) so that no pair of points has the distance square root of 3.
         x
        x x
       . . .
      x . . x
     x x . x x
    . . . . . .
   x . . x . . x
  x x . x x . x x
Continuation of this pattern will give the unique maximal solution for all n = 3j-1.
Triangle T(n, k) begins:
  1,  1;
  1,  3,   3,   1;
  1,  6,  12,   8;
  1, 10,  36,  55,   33,    9;
  1, 15,  87, 248,  378,  339,  187,   63,  12,   1;
  1, 21, 180, 820, 2190, 3606, 3716, 2340, 825, 125;
First row refers to n = 1.

Links

Crossrefs

Cf. A000108, A000326, A033536, A045943, A244506, A244507.
Cf. A000217 (column 2), A086274 (1/3 * column 3), A244501 (column 4), A244502 (column 5), A244503 (column 6).
Cf. A287195 (length of row n). - Eric W. Weisstein, May 21 2017
Cf. A287204 (row sums). - Eric W. Weisstein, May 21 2017

A244502 Number of ways to place 4 points on an n X n X n triangular grid so that no pair of them has distance sqrt(3).

Original entry on oeis.org

33, 378, 2190, 9110, 30300, 85563, 213293, 482085, 1006950, 1971185, 3655053, 6472533, 11017505, 18120840, 28919970, 44942618, 68206473, 101336700, 147703280, 211580280, 298329258, 414609113, 568614795, 770347395, 1031918240, 1367889723, 1795655703, 2335864415
Offset: 4

Views

Author

Heinrich Ludwig, Jun 29 2014

Keywords

Comments

sqrt(3) is the second closest (Euclidean) distance for a pair of points in a triangular grid. For illustration see A244500.

Links

Crossrefs

Cf. A086274, A244500, A244501, A244503.

Programs

  • Maple
    A244502:=n->`if`(n=4,33,1/384*n^8 + 1/96*n^7 - 13/64*n^6 + 5/48*n^5 + 875/128*n^4 - 2543/96*n^3 - 4141/96*n^2 + 3759/8*n - 837); seq(A244502(n), n=4..30); # Wesley Ivan Hurt, Jun 30 2014
  • Mathematica
    CoefficientList[Series[(10*x^9 - 30*x^8 + 130*x^6 - 333*x^5 + 444*x^4 - 236*x^3 + 24*x^2 - 81*x - 33)/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 30 2014 *)
  • PARI
    Vec(x^4*(10*x^9-30*x^8+130*x^6-333*x^5+444*x^4-236*x^3+24*x^2-81*x-33)/(x-1)^9 + O(x^100)) \\ Colin Barker, Jun 29 2014

Formula

a(n) = 1/384*n^8 + 1/96*n^7 - 13/64*n^6 + 5/48*n^5 + 875/128*n^4 - 2543/96*n^3 - 4141/96*n^2 + 3759/8*n - 837, for n >= 5.
G.f.: x^4*(10*x^9 - 30*x^8 + 130*x^6 - 333*x^5 + 444*x^4 - 236*x^3 + 24*x^2 - 81*x - 33) / (x - 1)^9. - Colin Barker, Jun 29 2014

A244503 Number of ways to place 5 points on an n X n X n triangular grid so that no pair of them has distance sqrt(3).

Original entry on oeis.org

9, 339, 3606, 24474, 121077, 475353, 1568712, 4524540, 11722134, 27828138, 61442460, 127616970, 251577939, 474068124, 858822579, 1502804622, 2549955858, 4209357693, 6778862319, 10675429650, 16473604089, 24953782251, 37162160802, 54484513344, 78736227726
Offset: 4

Views

Author

Heinrich Ludwig, Jun 29 2014

Keywords

Comments

sqrt(3) is the second closest (Euclidean) distance for a pair of points in a triangular grid. For illustration see A244500.
All elements of the sequence are multiples of 3.

Links

Crossrefs

Cf. A086274, A244500, A244501, A244502.

Programs

  • Mathematica
    CoefficientList[Series[-3*(5*x^13 -15*x^12 -26*x^11 +228*x^10 -584*x^9 +706*x^8 -162*x^7 -542*x^6 +766*x^5 -924*x^4 +656*x^3 +124*x^2 +80*x +3) / (x-1)^11, {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 03 2014 after Colin Barker *)

Formula

a(n) = 1/3840*n^10 + 1/768*n^9 - 13/384*n^8 - 7/384*n^7 + 1589/768*n^6 - 24619/3840*n^5 - 1561/32*n^4 + 20965/64*n^3 - 11101/240*n^2 - 85143/20*n + 9711 for n >= 7.
G.f.: -3*x^4*(5*x^13 - 15*x^12 - 26*x^11 + 228*x^10 - 584*x^9 + 706*x^8 - 162*x^7 - 542*x^6 + 766*x^5 - 924*x^4 + 656*x^3 + 124*x^2 + 80*x + 3) / (x - 1)^11. - Colin Barker, Jun 29 2014
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