OFFSET
0,6
COMMENTS
Skew triangle associated with the Fibonacci numbers.
LINKS
H. Fuks and J. M. G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.
FORMULA
Sum_{k=0..n} T(n,k) = A011782(n).
Sum_{n>=k} T(n,k) = A001333(k).
T(n,k) = 0 if k < 0 or if k > n, T(0,0) = 1, T(2,1) = 0, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2).
T(n,n) = Fibonacci(n+1) = A000045(n+1).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A133592(n), A133594(n), A133642(n), A133646(n), A133678(n), A133679(n), A133680(n), A133681(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Jan 03 2008
G.f.: (1-y*x^2)/(1-y*x-y*(y+1)*x^2). - Philippe Deléham, Nov 26 2011
EXAMPLE
Triangle begins:
1;
0, 1;
0, 0, 2;
0, 0, 1, 3;
0, 0, 0, 3, 5;
0, 0, 0, 1, 7, 8;
0, 0, 0, 0, 4, 15, 13;
0, 0, 0, 0, 1, 12, 30, 21;
0, 0, 0, 0, 0, 5, 31, 58, 34;
0, 0, 0, 0, 0, 1, 18, 73, 109, 55;
0, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89;
0, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144;
0, 0, 0, 0, 0, 0, 0, 7, 85, 361, 707, 655, 233;
MATHEMATICA
T[0, 0] = T[1, 1] = 1; T[_, 0] = T[_, 1] = 0; T[n_, n_] := Fibonacci[n+1]; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]; T[_, _] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Oct 25 2006
STATUS
approved