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

0,6

Comments

Skew triangle associated with the Fibonacci numbers.

Links

Table of n, a(n) for n=0..88.

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

Cf. A055830 (another version).

Sequence in context: A350530 A258850 A182114 * A374766 A116489 A166373

Adjacent sequences: A122947 A122948 A122949 * A122951 A122952 A122953

Keyword

nonn,tabl

Author

Philippe Deléham, Oct 25 2006

Status

approved