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Search: id:A129713
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| A129713 |
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Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 1's (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword. |
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+0
1
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| 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 8, 5, 3, 2, 1, 1, 1, 13, 8, 5, 3, 2, 1, 1, 1, 21, 13, 8, 5, 3, 2, 1, 1, 1, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 233, 144
(list; table; graph; listen)
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OFFSET
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0,7
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COMMENT
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Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), 0<=k<=n)=F(n+3)-2=A001911(n).
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FORMULA
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T(n,k)=F(n-k) if k<=n-2, T(n,n-1)=T(n,n)=1, where F(j) are the Fibonacci numbers (F(0)=0, F(1)=1). G.f.=G(t,z)=(1-z^2)/[(1-z-z^2)(1-tz)].
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EXAMPLE
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T(6,2)=3 because we have 110110, 110111, 110101.
Triangle starts:
1;
1,2;
1,1,1;
2,1,1,1;
3,2,1,1,1;
5,3,2,1,1,1;
8,5,3,2,1,1,1;
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MAPLE
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with(combinat): T:=proc(n, k) if k<=n-2 then fibonacci(n-k) elif k=n-1 or k=n then 1 else 0 fi end: for n from 0 to 15 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000045, A001911.
Sequence in context: A123320 A054123 A119269 this_sequence A096669 A096591 A115568
Adjacent sequences: A129710 A129711 A129712 this_sequence A129714 A129715 A129716
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 12 2007
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