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 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. 19
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A055830 (another version). Sequence in context: A350530 A258850 A182114 * A116489 A166373 A202451 Adjacent sequences: A122947 A122948 A122949 * A122951 A122952 A122953 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Oct 25 2006 STATUS approved

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Last modified March 22 08:23 EDT 2023. Contains 361419 sequences. (Running on oeis4.)