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A122950
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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.
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19
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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
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OFFSET
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0,6
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COMMENTS
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Skew triangle associated with the Fibonacci numbers.
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A055830 (another version).
Sequence in context: A350530 A258850 A182114 * A116489 A166373 A202451
Adjacent sequences: A122947 A122948 A122949 * A122951 A122952 A122953
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe Deléham, Oct 25 2006
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STATUS
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approved
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