A240439
Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 105, 114, 39, 3, 1, 15, 105, 420, 969, 1194, 654, 102, 3, 1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15, 1, 28, 378, 3150, 17415, 64776, 159528, 250233, 234609, 119259, 28395, 2613, 69, 1, 36, 630, 6930
Offset: 1
The triangle begins:
1, 1;
1, 3, 3;
1, 6, 15, 15, 3;
1, 10, 45, 105, 114, 39, 3;
1, 15, 105, 420, 969, 1194, 654, 102, 3;
1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15;
There are T(5, 8) = 3 ways to place 8 points (x) on a triangular grid of side 5 under the conditions mentioned above:
. x x
x x x . . x
x . x x . . . . x
x . . x x . . . . . . x
x . . . x . x x x x x x x x .
A240440
Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.
Original entry on oeis.org
0, 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, 45045, 75075, 120120, 185640, 278460, 406980, 581400, 813960, 1119195, 1514205, 2018940, 2656500, 3453450, 4440150, 5651100, 7125300, 8906625, 11044215, 13592880, 16613520, 20173560, 24347400, 29216880
Offset: 1
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=5 in the last equation on page 3.
- Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
If one of the initial zeros is omitted, this is a row of the array in
A129533.
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[(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015
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A240440:=n->(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48; seq(A240440(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014
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Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)
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Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014
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vector(100,n,(n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015
A240442
Number of ways to place 5 points on a triangular grid of side n so that no three of these points are vertices of an equilateral triangle of any orientation.
Original entry on oeis.org
0, 39, 1194, 11259, 64776, 275805, 957516, 2859768, 7606821, 18444537, 41458599, 87464157, 174846963, 333687378, 611613150, 1081890447
Offset: 3
A243143
Number of inequivalent (mod D_3) ways to place 4 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.
Original entry on oeis.org
1, 22, 170, 816, 2947, 8765, 22703, 52823, 113042, 225817, 426299, 766905, 1324282, 2206478, 3563770, 5599258, 8584775, 12875840, 18934040, 27347390, 38860741, 54402707, 75125825, 102441321, 138070912, 184090795, 242997153, 317760863, 411908932, 529591532, 675681764
Offset: 3
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-7,-5,23,-19,-7,27,-27,7,19,-23,5,7,-5,1).
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Drop[CoefficientList[Series[-x^3*(3*x^10 - 10*x^9 + 19*x^8 - 13*x^7 + 102*x^6 + 105*x^5 + 144*x^4 + 125*x^3 + 67*x^2 + 17*x + 1) / ((x-1)^9*(x+1)^4*(x^2+1)), {x, 0, 40}], x],3] (* Vaclav Kotesovec, May 31 2014 after Colin Barker *)
Showing 1-4 of 4 results.
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