A244500 -id:A244500 - cp's OEIS frontend

A244501 Number of ways to place 3 points on an n X n X n triangular grid so that no pair of them has distance sqrt(3).

1, 8, 55, 248, 820, 2212, 5163, 10815, 20833, 37540, 64067, 104518, 164150, 249568, 368935, 532197, 751323, 1040560, 1416703, 1899380, 2511352, 3278828, 4231795, 5404363, 6835125, 8567532, 10650283, 13137730, 16090298, 19574920, 23665487, 28443313, 33997615

Offset: 2

Heinrich Ludwig, Jun 29 2014

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

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A086274, A244500, A244502, A244503.

CoefficientList[Series[-(6*x^7-17*x^6+14*x^5-6*x^4-4*x^3+20*x^2+x+1) / (x-1)^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 03 2014 after Colin Barker *)

Vec(-x^2*(6*x^7-17*x^6+14*x^5-6*x^4-4*x^3+20*x^2+x+1)/(x-1)^7 + O(x^100)) \\ Colin Barker, Jun 29 2014

a(n) = 1/48*n^6 + 1/16*n^5 - 13/16*n^4 + 61/48*n^3 + 247/24*n^2 - 293/6*n + 6 for n >= 3.

G.f.: -x^2*(6*x^7 - 17*x^6 + 14*x^5 - 6*x^4 - 4*x^3 + 20*x^2 + x + 1) / (x-1)^7. - Colin Barker, Jun 29 2014

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

Heinrich Ludwig, Jun 29 2014

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

Heinrich Ludwig, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A086274, A244500, A244501, A244503.

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

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 *)

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

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

Heinrich Ludwig, Jun 29 2014

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.

Heinrich Ludwig, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A086274, A244500, A244501, A244502.

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 *)

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

A244506 Number of ways to place the maximal number of points that can be placed on a j X j X j triangular grid, j=3n-2, so that no pair of them has distance sqrt(3).

Original entry on oeis.org

1, 9, 196, 6084, 219024, 8450649, 338265664, 13840346025, 574510941225, 24093764931600

Offset: 1

Heinrich Ludwig, Jul 11 2014

(1) All a(n) are square numbers. The sequence of their roots is A244507.
(2) On a j X j X j grid, j = 3n-2, the maximal number of points that can be placed is the pentagonal number A000326(n).
(3) On a maximally occupied grid, the following grid points "X" are always occupied (example for j = 7, for other j's expand this pattern):
              X
             . .
            . . .
           x . . X
          . . . . .
         . . . . . .
        X . . X . . X
(4) For j X j X j grids, j = 3n, the corresponding numbers are cubes of Catalan numbers, A033536(n). For j = 3n-1, the corresponding numbers are always 1, A000012(n).

Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Cf. A000326, A244507, A244500.

Cf. A297537 (maximum independent vertex sets for n-triangular honeycomb acute knight graph).

a(10) from Heinrich Ludwig, Aug 17 2014

A244507 Square roots of A244506.

Original entry on oeis.org

1, 3, 14, 78, 468, 2907, 18392, 117645, 757965, 4908540

Offset: 1

Heinrich Ludwig, Jul 11 2014

A244506 gives the number of ways to place the maximal number of points that can be placed on a j X j X j triangular grid, j = 3n-2, so that no pair of them has distance sqrt(3). These numbers are all square numbers. A244507 gives the roots of them.

Cf. A244506, A244500.

a(10) from Heinrich Ludwig, Aug 17 2014

A287195 Independence and clique covering number of the n-triangular honeycomb acute knight graph.

Original entry on oeis.org

1, 3, 3, 5, 9, 9, 12, 18, 18, 22, 30, 30, 35, 45, 45, 51, 63, 63, 70, 84, 84, 92, 108, 108, 117, 135, 135, 145, 165, 165, 176, 198, 198, 210, 234, 234, 247, 273, 273, 287, 315, 315, 330, 360, 360, 376, 408, 408, 425, 459, 459, 477, 513, 513, 532, 570, 570

Offset: 1

Eric W. Weisstein, May 21 2017

a(n) is also the length of row n in A244500.

Colin Barker, Table of n, a(n) for n = 1..1000
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Independence Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A244500.

Cf. A066377, A092353.

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 3, 3, 5, 9, 9, 12}, 50]
Table[1/18 ((n + 3) (3 n + 2) - 2 (n + 3) Cos[2 n Pi/3] - 2 Sqrt[3] (n + 1) Sin[2 n Pi/3]), {n, 50}]
Table[Piecewise[{{n (n + 3), Mod[n, 3] == 0}, {(n + 1) (n + 2), Mod[n, 3] == 1}, {(n + 1) (n + 4), Mod[n, 3] == 2}}]/6, {n, 50}]

Vec(x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jul 15 2017

From Colin Barker, Jul 15 2017: (Start)
G.f.: x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7. (End)
From Ridouane Oudra, Jun 23 2024: (Start)
a(n) = Sum_{i=1..n+3} (i mod 3)*floor(i/3);
a(n) = (1/2)*(n^2 + n + (n^2 - 5*n)*t -(6*n - 9)*t^2 + 9*t^3), where t = floor(n/3);
a(n) = A066377(n+1) - A092353(n). (End)
E.g.f.: exp(-x/2)*(exp(3*x/2)*(6 + 14*x + 3*x^2) - 2*(3 + x)*cos(sqrt(3)*x/2) - 2*sqrt(3)*(1 - x)*sin(sqrt(3)*x/2))/18. - Stefano Spezia, Jun 23 2024

A287204 Number of independent vertex sets and vertex covers in the n-triangular honeycomb acute knight graph.

Original entry on oeis.org

2, 8, 27, 144, 1331, 13824, 208575, 4741632, 140608000, 5870657088, 350253538649, 28635764824584, 3270761581689147, 524882236329014264, 117025388905265182783, 36405238728128712859232, 15833111238624255435917875, 9599917578598545475054063616

Offset: 1

Eric W. Weisstein, May 21 2017

nonn

Also row sums of A244500.

Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A244500 (coefficients of independence polynomial of triangular honeycomb acute knight graphs).

Cf. A287195, A289902, A287230, A287229, A027740.

a(15)-a(18) from Andrew Howroyd, Jul 17 2017

cp's OEIS Frontend

A244501 Number of ways to place 3 points on an n X n X n triangular grid so that no pair of them has distance sqrt(3).

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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).

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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).

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A244506 Number of ways to place the maximal number of points that can be placed on a j X j X j triangular grid, j=3n-2, so that no pair of them has distance sqrt(3).

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A244507 Square roots of A244506.

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A287195 Independence and clique covering number of the n-triangular honeycomb acute knight graph.

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A287204 Number of independent vertex sets and vertex covers in the n-triangular honeycomb acute knight graph.

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