A239572
Triangle T(n, k) = Numbers of non-equivalent (mod D_3) ways to place k points on a triangular grid of side n so that no two of them are adjacent. Triangle read by rows.
Original entry on oeis.org
1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 16, 32, 24, 7, 1, 5, 32, 113, 200, 176, 66, 6, 7, 60, 329, 1053, 1976, 2096, 1162, 302, 34, 2, 8, 100, 790, 3932, 12565, 25676, 32963, 25638, 11294, 2493, 222, 7, 10, 160, 1702, 11988, 57275, 187984, 425329, 658608, 684671, 462519
Offset: 1
Triangle begins
1;
1;
2, 2, 1;
3, 6, 6, 1;
4, 16, 32, 24, 7, 1;
5, 32, 113, 200, 176, 66, 6;
7, 60, 329, 1053, 1976, 2096, 1162, 302, 34, 2;
8, 100, 790, 3932, 12565, 25676, 32963, 25638, 11294, 2493, 222, 7;
A239569
Number of ways to place 3 points on a triangular grid of side n so that no two of them are adjacent.
Original entry on oeis.org
0, 1, 21, 151, 615, 1845, 4571, 9926, 19566, 35805, 61765, 101541, 160381, 244881, 363195, 525260, 743036, 1030761, 1405221, 1886035, 2495955, 3261181, 4211691, 5381586, 6809450, 8538725, 10618101, 13101921, 16050601, 19531065, 23617195, 28390296, 33939576
Offset: 2
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[(n^2-3*n+2)*(n^4+6*n^3-23*n^2-92*n+264)/48: n in [2..40]]; // Vincenzo Librandi, Mar 23 2014
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CoefficientList[Series[- x (11 x^4 - 36 x^3 + 25 x^2 + 14 x + 1)/(x - 1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,21,151,615,1845,4571},50] (* Harvey P. Dale, Aug 08 2023 *)
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concat(0, Vec(-x^3*(11*x^4-36*x^3+25*x^2+14*x+1)/(x-1)^7 + O(x^100))) \\ Colin Barker, Mar 22 2014
A239574
Number of non-equivalent (mod D_3) ways to place 4 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.
Original entry on oeis.org
0, 1, 24, 200, 1053, 3932, 11988, 31298, 73046, 155880, 310046, 581414, 1038634, 1779531, 2942114, 4714412, 7350595, 11184786, 16654116, 24317554, 34886940, 49252544, 68523846, 94062350, 127534794, 170954603, 226748678, 297809946, 387580007, 500113190, 640178710
Offset: 3
There is a(4) = 1 way to place 4 points on a triangular grid of side n = 4:
X
. .
. X .
X . . X
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1)
-
Drop[CoefficientList[Series[x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3), {x, 0, 20}], x],3] (* Vaclav Kotesovec, Mar 29 2014 *)
Table[(n^8+4*n^7-78*n^6-104*n^5+2556*n^4-3152*n^3-27280*n^2+89664*n-78336)/2304 + If[Mod[n,2]==1,(28*n^3-54*n^2-160*n+129)/768,0] + If[Mod[n,3]==1,(n^2+n-14)/18,0],{n,3,20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 29 2014 *)
A239575
Number of non-equivalent (mod D_3) ways to place 5 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.
Original entry on oeis.org
0, 0, 7, 176, 1976, 12565, 57275, 207018, 634166, 1711262, 4181915, 9428657, 19892816, 39684027, 75473209, 137721045, 242391212, 413215132, 684733527, 1106194950, 1746637600, 2701244609, 4099429895, 6114748948, 8977257362, 12988406970, 18539308619, 26132434991
Offset: 3
There are a(5) = 7 non-equivalent ways to place 5 points (x) on a triangular grid of side 5. These are:
x x . x
. . . . . . . .
x . x x . x x . x . x .
. . . . . . . . . . . . . . . .
x . . . x . x . x . x . x . x x . x . x
.
x x x
. . . . . .
. x . . x . x . x
x . . x x . . . . . . .
. . x . . . . x . x x . . x .
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1)
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Table[(n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + (1-(-1)^n)/2*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536,{n,3,20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 31 2014 *)
Drop[CoefficientList[Series[x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11*(1+x)^5)), {x, 0, 20}], x], 3] (* Vaclav Kotesovec, Mar 31 2014 *)
A279446
Number of non-equivalent (mod D_3) ways to place 6 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.
Original entry on oeis.org
0, 0, 1, 66, 2096, 25676, 187984, 983172, 4073312, 14196011, 43309138, 118818916, 298926225, 699619679, 1540212590, 3217045155, 6419240369, 12304959047, 22763742133, 40797668697, 71065355815, 120643462032, 200077436639, 324808463585, 517088445952, 808515893580
Offset: 3
There is a(5) = 1 way to place 6 points on a triangular grid of side n = 5:
X
. .
X . X
. . . .
X . X . X
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (4,0,-17,8,36,-7,-68,-18,113,52,-126,-92,92,126,-52,-113,18,68,7,-36,-8,17,0,-4,1).
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Table[Boole[n > 4] ((n^12 + 6 n^11 - 195 n^10 - 670 n^9 + 17455 n^8 + 13426 n^7 - 835256 n^6 + 1246240 n^5 + 19563664 n^4 - 68181792 n^3 - 131524224 n^2 + 969500160 n - 1298903040)/276480 + Boole[OddQ@ n] (162 n^5 - 715 n^4 - 4480 n^3 + 21955 n^2 + 1108 n - 41685)/30720 + Boole[Mod[n, 3] == 1] (n^2 + n - 25)/27), {n, 3, 28}] (* Michael De Vlieger, Feb 26 2017 *)
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concat(vector(2), Vec(x^5*(1 + 62*x + 1832*x^2 + 17309*x^3 + 86394*x^4 + 266304*x^5 + 557979*x^6 + 818157*x^7 + 829988*x^8 + 519203*x^9 + 94134*x^10 - 150065*x^11 - 123434*x^12 + 7445*x^13 + 64052*x^14 + 29943*x^15 - 11247*x^16 - 15803*x^17 - 3012*x^18 + 3100*x^19 + 1722*x^20 - 15*x^21 - 233*x^22 - 56*x^23) / ((1 - x)^13*(1 + x)^6*(1 + x + x^2)^3) + O(x^30))) \\ Colin Barker, Feb 26 2017
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