A243209
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 with sides parallel to the grid.
Original entry on oeis.org
1, 25, 186, 881, 3146, 9264, 23810, 55058, 117205, 233135, 438544, 786541, 1354696, 2252202, 3630684, 5694984, 8718963, 13060515, 19184110, 27681103, 39300096, 54974216, 75861038, 103377456, 139251749, 185567453, 244828780, 320015885, 414665890, 532940080
Offset: 3
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (3,0,-6,0,6,8,-12,-9,13,6,-6,-13,9,12,-8,-6,0,6,0,-3,1).
-
Drop[CoefficientList[Series[-x^3*(1 + 22*x + 111*x^2 + 329*x^3 + 653*x^4 + 936*x^5 + 1146*x^6 + 1200*x^7 + 1150*x^8 + 900*x^9 + 650*x^10 + 286*x^11 + 131*x^12 + 28*x^13 + 19*x^14 - 5*x^15 + 3*x^16) / ((-1+x)^9 * (1+x)^4 * (1-x+x^2) * (1+x+x^2)^3), {x, 0, 40}], x],3] (* Vaclav Kotesovec, Jun 02 2014 *)
A243211
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 with sides parallel to the grid. Triangle read by rows.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 107, 128, 63, 10, 1, 15, 105, 428, 1062, 1566, 1276, 507, 69, 1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1, 1, 28, 378, 3198, 18591, 77124, 231090, 498097, 759117, 792942, 540361, 222597, 49053
Offset: 1
The triangle begins:
1, 1;
1, 3, 3;
1, 6, 15, 15, 3;
1, 10, 45, 107, 128, 63, 10,
1, 15, 105, 428, 1062, 1566, 1276, 507, 69,
1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1;
...
There is T(6, 12) = 1 way to place 12 points (x) on the grid obeying the rule in the definition of the sequence:
.
x x
x . x
x . . x
x . . . x
. x x x x .
Cf.
A227308,
A243207,
A084546,
A234251,
A239567,
A240439,
A194136,
A000217 (column 2),
A050534 (column 3),
A243212 (column 4),
A243213 (column 5),
A243214 (column 6).
A243212
Number of ways to place 3 points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.
Original entry on oeis.org
0, 15, 107, 428, 1282, 3198, 7022, 14020, 26000, 45445, 75665, 120960, 186802, 280028, 409052, 584088, 817392, 1123515, 1519575, 2025540, 2664530, 3463130, 4451722, 5664828, 7141472, 8925553, 11066237, 13618360, 16642850, 20207160, 24385720, 29260400, 34920992
Offset: 2
- Heinrich Ludwig, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).
-
I:=[0,15,107,428,1282,3198,7022,14020]; [n le 8 select I[n] else 6*Self(n-1)-14*Self(n-2)+14*Self(n-3)-14*Self(n-5)+14*Self(n-6)-6*Self(n-7)+Self(n-8): n in [1..40]]; // Vincenzo Librandi, Jun 23 2015
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Table[Binomial[n (n + 1)/2, 3] - Floor[(n - 1) (n + 1) (2 n - 1)/8], {n, 2, 40}] (* Vincenzo Librandi, Jun 23 2015 *)
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concat(0, Vec(-x^3*(2*x^3-4*x^2+17*x+15)/((x-1)^7*(x+1)) + O(x^100))) \\ Colin Barker, Jun 09 2014
A243214
Number of ways to place 5 points on a triangular grid of side n, so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.
Original entry on oeis.org
0, 63, 1566, 13971, 77124, 319206, 1083723, 3181401, 8344854, 20006349, 44548227, 93248628, 185176866, 351410664, 640972980, 1129067352, 1928196432, 3203016813
Offset: 3
Showing 1-4 of 4 results.
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