login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A157897 Triangle read by rows, 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. 4

%I

%S 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,

%T 3,3,0,1,7,15,16,21,12,6,3,0,1,8,21,27,35,30,14,12,0,1

%N Triangle read by rows, 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.

%C Sum of n-th row = A000073(n+2). - _Reinhard Zumkeller_, Jun 25 2009

%C 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

%C 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

%H K. Edwards, <a href="http://www.fq.math.ca/Papers1/46_47-1/Edwards11-08.pdf">A Pascal-like triangle related to the tribonacci numbers</a>, Fib. Q., 46/47 (2008/2009), 18-25.

%H Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.

%F 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.

%e First few rows of the triangle are:

%e 1;

%e 1, 0;

%e 1, 1, 0;

%e 1, 2, 0, 1;

%e 1, 3, 1, 2, 0;

%e 1, 4, 3, 3, 2, 0;

%e 1, 5, 6, 5, 6, 0, 1;

%e 1, 6, 10, 9, 12, 3, 3, 0;

%e 1, 7, 15, 16, 21, 12, 6, 3, 0;

%e 1, 8, 21, 27, 35, 30, 14, 12, 0, 1;

%e ...

%e T(9,3) = 27 = T(8,3) + T(7,2) + T(6,0) = 16 + 10 + 1.

%t 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,0,9},{k,0,n}]] (* _Michael A. Allen_, Apr 28 2019 *)

%Y Cf. A120415, A006498.

%Y Other triangles related to tiling using fences: A059259, A123521, A335964.

%K nonn,tabl

%O 0,8

%A _Gary W. Adamson_, Mar 08 2009

%E Name clarified by _Michael A. Allen_, Apr 28 2019

%E Definition improved by _Michael A. Allen_, Mar 11 2021

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 19 12:41 EDT 2021. Contains 345129 sequences. (Running on oeis4.)