OFFSET
0,8
COMMENTS
T(n, k) is the number of tilings of an n-board that use k (1/2, 1)-fences and n-k squares. A (1/2, 1)-fence is a tile composed of two pieces of width 1/2 separated by a gap of width 1. (Result proved in paper by K. Edwards - see the links section.) - Michael A. Allen, Apr 28 2019
T(n, k) is the (n, n-k)-th entry in the (1/(1-x^3), x*(1+x)/(1-x^3)) Riordan array. - Michael A. Allen, Mar 11 2021
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fib. Q., 46/47 (2008/2009), 18-25.
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, J. Int. Seq. 24 (2021) Article 21.3.8.
FORMULA
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n<k,k) = 0.
Sum_{k=0..n} T(n, k) = A000073(n+2). - Reinhard Zumkeller, Jun 25 2009
From G. C. Greubel, Sep 01 2022: (Start)
T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-3), with T(n, 0) = 1.
T(n, n) = A079978(n).
T(n, n-1) = A087508(n), n >= 1.
T(n, 1) = A001477(n-1).
T(n, 2) = A161680(n-2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A120415(n). (End)
EXAMPLE
First few rows of the triangle are:
1;
1, 0;
1, 1, 0;
1, 2, 0, 1;
1, 3, 1, 2, 0;
1, 4, 3, 3, 2, 0;
1, 5, 6, 5, 6, 0, 1;
1, 6, 10, 9, 12, 3, 3, 0;
1, 7, 15, 16, 21, 12, 6, 3, 0;
1, 8, 21, 27, 35, 30, 14, 12, 0, 1;
...
T(9,3) = 27 = T(8,3) + T(7,2) + T(6,0) = 16 + 10 + 1.
MATHEMATICA
T[n_, k_]:= If[n<k || k<0, 0, T[n-1, k]+T[n-2, k-1]+T[n-3, k-3]+KroneckerDelta[n, k, 0]];
Flatten[Table[T[n, k], {n, 0, 14}, {k, 0, n}]] (* Michael A. Allen, Apr 28 2019 *)
PROG
(Magma)
function T(n, k) // T = A157897
if k lt 0 or k gt n then return 0;
elif k eq 0 then return 1;
else return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 01 2022
(SageMath)
def T(n, k): # T = A157897
if (k<0 or k>n): return 0
elif (k==0): return 1
else: return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3)
flatten([[T(n, k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Sep 01 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Mar 08 2009
EXTENSIONS
Name clarified by Michael A. Allen, Apr 28 2019
Definition improved by Michael A. Allen, Mar 11 2021
STATUS
approved