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Search: id:A092921
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| A092921 |
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Array F(k,n) read by antidiagonals: k-generalized Fibonacci numbers. |
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13
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| 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 2, 1, 1, 0, 1, 5, 4, 2, 1, 1, 0, 1, 8, 7, 4, 2, 1, 1, 0, 1, 13, 13, 8, 4, 2, 1, 1, 0, 1, 21, 24, 15, 8, 4, 2, 1, 1, 0, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 0, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 0, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1
(list; table; graph; listen)
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OFFSET
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0,12
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COMMENT
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For all k>=1, the k-generalized Fibonacci number F(k,n) satisfies the recurrence obtained by adding more terms to the recurrence of the Fibonacci numbers.
The number of tilings of an 1 X n rectangle with tiles of size 1 X 1, 1 X 2, ..., 1 X k is F(k,n).
T(k,n) is the number of 0-balanced ordered trees with n edges and height k (height is the number of edges from root to a leaf). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 19 2007
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REFERENCES
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Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
H. Gabai, Generalized Fibonacci k-sequences, Fib. Quart., 8 (1970), 31-38.
R. Kemp, Balanced ordered trees, Random Structures and Alg., 5 (1994), pp. 99-121.
E. P. Miles jr., Generalized Fibonacci numbers and associated matrices, The Amer. Math. Monthly, 67 (1960) 745-752.
M. D. Miller, On generalized Fibonacci numbers, The Amer. Math. Monthly, 78 (1971) 1108-1109.
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LINKS
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E. S. Egge, Restricted permutations related to Fibonacci numbers....
E. S. Egge, Restricted 3412-Avoiding Involutions
E. S. Egge and T. Mansour, Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers.
E. S. Egge and T. Mansour, 231-avoiding involutioms and Fibonacci numbers.
A. Flaxman, A. W. Harrow and G. B. Sorkin, Strings with maximally many distinct subsequences and substrings
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FORMULA
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F(k, n)=F(k, n-1)+F(k, n-2)+...+F(k, n-k); F(k, 1)=1 and for n<=0, F(k, n)=0.
G.f.: x/[1-sum(i=0..k, x^i)].
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PROGRAM
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(PARI) F(k, n)=if(n<2, if(n<1, 0, 1), sum(i=1, k, F(k, n-i)))
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CROSSREFS
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Columns converge to 2^(n-2).
Rows 1-8 are (shifted) A057427, A000045, A000073, A000078, A001591, A001592, A066178, A079262.
Essentially a reflected version of A048887. See A048004 and A126198 for closely related arrays.
Sequence in context: A105806 A129501 A158511 this_sequence A029387 A070878 A060959
Adjacent sequences: A092918 A092919 A092920 this_sequence A092922 A092923 A092924
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, Apr 17 2004
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