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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<0,k) = T(n<k,k) = 0. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..77.

Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.

FORMULA

T(n,k) = A059259(n-k,k).

T(n,2) = binomial(n-4,2) + n - 3.

T(n,3) = binomial(n-6,3) + 2*binomial(n-5,2).

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 || n<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, 9}, {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.

Cf. A006498.

Sequence in context: A076833 A071676 A319933 * A301570 A301567 A115363

Adjacent sequences:  A335961 A335962 A335963 * A335965 A335966 A335967

KEYWORD

easy,nonn,tabl

AUTHOR

Michael A. Allen, Jul 01 2020

STATUS

approved

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Last modified June 22 16:01 EDT 2021. Contains 345386 sequences. (Running on oeis4.)