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URL: https://oeis.org/A335964

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A335964
Triangle read by rows, T(n,k) = T(n-1,k) + T(n-3,k-1) + T(n-4,k-2) + delta(n,0)*delta(k,0), T(n,k<0) = T(n<k,k) = 0.
11
1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 0, 1, 4, 4, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 1, 6, 11, 6, 1, 0, 0, 0, 0, 1, 7, 16, 13, 3, 0, 0, 0, 0, 0, 1, 8, 22, 24, 9, 0, 0, 0, 0, 0, 0, 1, 9, 29, 40, 22, 3, 0, 0, 0, 0, 0, 0
OFFSET
0,12
COMMENTS
T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1,1)-fence tiles and n-2k square tiles. A (w,g)-fence tile is composed of two tiles of width w separated by a gap of width g.
Sum of n-th row = A006498(n).
T(2*j+r,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x) = f(n,x) + x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0. - Michael A. Allen, Oct 02 2021
LINKS
John Konvalina, On the number of combinations without unit separation., Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table I.
FORMULA
T(n,k) = A059259(n-k,k).
From Michael A. Allen, Oct 02 2021: (Start)
G.f.: 1/((1 + x^2*y)(1 - x - x^2*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the expansion of the g.f.
T(n,0) = 1.
T(n,1) = n-2 for n>1.
T(n,2) = binomial(n-4,2) + n - 3 for n>3.
T(n,3) = binomial(n-6,3) + 2*binomial(n-5,2) for n>5.
T(4*m-3,2*m-2) = T(4*m-1,2*m-1) = m for m>0.
T(2*n+1,n-k) = A158909(n,k). (End)
T(n,k) = A348445(n-2,k) for n>1.
EXAMPLE
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 1, 0, 0;
1, 2, 1, 0, 0;
1, 3, 2, 0, 0, 0;
1, 4, 4, 0, 0, 0, 0;
1, 5, 7, 2, 0, 0, 0, 0;
1, 6, 11, 6, 1, 0, 0, 0, 0;
1, 7, 16, 13, 3, 0, 0, 0, 0, 0;
1, 8, 22, 24, 9, 0, 0, 0, 0, 0, 0;
1, 9, 29, 40, 22, 3, 0, 0, 0, 0, 0, 0;
...
MATHEMATICA
T[n_, k_]:=If[n<k || k<0, 0, T[n-1, k] + T[n-3, k-1] + T[n-4, k-2] + KroneckerDelta[n, k, 0]]; Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, n}]]
(* or via the g.f.: *)
Flatten[Table[CoefficientList[Series[1/((1+x^2*y)(1 - x - x^2*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n+1, k+1]], {n, 0, 11}, {k, 0, n}]]
PROG
(PARI) TT(n, k) = if (n<k, 0, if((n==0) || (k==0), 1, if(k==n, (1+(-1)^n)/2, TT(n-1, k)+TT(n-1, k-1)))) \\ A059259
T(n, k) = TT(n-k, k);
\\ matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jul 18 2020
CROSSREFS
Other triangles related to tiling using fences: A059259, A123521, A157897, A158909.
Cf. A006498 (row sums), A011973, A348445.
Sequence in context: A076833 A071676 A319933 * A301570 A301567 A115363
KEYWORD
easy,nonn,tabl
AUTHOR
Michael A. Allen, Jul 01 2020
STATUS
approved