A122950 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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

Offset: 0

Author

Philippe Deléham, Oct 25 2006

Keywords

• nonn
• tabl

Comments

Skew triangle associated with the Fibonacci numbers.

Examples

			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;
		

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.

Crossrefs

Cf. A055830 (another version).

Programs

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 *)

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