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A371967
Irregular triangle T(r,w) read by rows: number of ways of placing w non-attacking wazirs on a 3 X r board.
6
1, 1, 3, 1, 1, 6, 8, 2, 1, 9, 24, 22, 6, 1, 1, 12, 49, 84, 61, 18, 2, 1, 15, 83, 215, 276, 174, 53, 9, 1, 1, 18, 126, 442, 840, 880, 504, 158, 28, 2, 1, 21, 178, 792, 2023, 3063, 2763, 1478, 472, 93, 12, 1, 1, 24, 239, 1292, 4176, 8406, 10692, 8604, 4374, 1416, 297, 38, 2, 1, 27, 309
OFFSET
0,3
LINKS
Alois P. Heinz, Rows r = 0..165 (first 17 rows from R. J. Mathar)
Jacob A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869 [math.CO] (2014) Table 2.
Wikipedia, Wazir (chess)
FORMULA
T(r,0) = 1.
T(r,1) = 3*r.
T(r,2) = A064225(r-1).
T(r,3) = A172229(r).
T(r,4) = 27*r^4/8 -117*r^3/4 +829*r^2/8 -715*r/4 +126. [Siehler Table 3]
T(3,w) = A232833(3,w).
G.f.: (1+x*y) *(1 +x*y +x*y^2 -x^2*y^3)/(1 -x -x*y -x^2*y^3 -2*x^2*y -3*x^2*y^2 -x^3*y^2 +x^3*y^4 +x^4*y^4). - R. J. Mathar, Apr 21 2024
EXAMPLE
The triangle starts with r>=0 rows and w>=0 wazirs as
1 ;
1 3 1 ;
1 6 8 2 ;
1 9 24 22 6 1 ;
1 12 49 84 61 18 2 ;
1 15 83 215 276 174 53 9 1 ;
1 18 126 442 840 880 504 158 28 2 ;
1 21 178 792 2023 3063 2763 1478 472 93 12 1 ;
1 24 239 1292 4176 8406 10692 8604 4374 1416 297 38 2 ;
1 27 309 1969 7731 19591 32716 36257 26674 13035 4264 945 142 15 1 ;
...
MAPLE
b:= proc(n, l) option remember; `if`(n=0, 1,
add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10); # Alois P. Heinz, Apr 14 2024
MATHEMATICA
b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[BitAnd[j, l] > 0, 0, Expand[b[n - 1, j]*x^DigitCount[j, 2, 1]]], {j, {0, 1, 2, 4, 5}}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 05 2024, after Alois P. Heinz *)
CROSSREFS
Cf. A051736 (row sums), A035607 (on 2Xr board), A011973 (on 1Xr board), A232833 (on rXr board).
T(n,n) gives A371978.
Row maxima give A371979.
Cf. A007494.
Sequence in context: A370757 A158359 A046716 * A202605 A298636 A123354
KEYWORD
nonn,tabf
AUTHOR
R. J. Mathar, Apr 14 2024
STATUS
approved