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Search: id:A129711
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| A129711 |
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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 01's (0<=k<=floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword. |
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+0
1
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| 1, 2, 2, 1, 3, 2, 5, 2, 1, 8, 3, 2, 13, 5, 2, 1, 21, 8, 3, 2, 34, 13, 5, 2, 1, 55, 21, 8, 3, 2, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 2, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 2, 610, 233, 89, 34, 13, 5, 2, 1, 987, 377, 144, 55, 21, 8, 3, 2, 1597, 610, 233, 89, 34, 13, 5, 2, 1
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=F(n+1)-1=A000071(n+1).
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FORMULA
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T(n,k)=F(n-2k+1) if 2k+1<n, where F(j) are the Fibonacci numbers (F(0)=0, F(1)=1); T(2k+1,k)=2; T(n,k)=0 if 2k>n. G.f.=G(t,z)=(1+z)(1-z^2)/[(1-z-z^2)(1-tz^2)].
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EXAMPLE
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T(7,2)=3 because we have 0101110, 0101111, and 0101101.
Triangle starts:
1;
2;
2,1;
3,2;
5,2,1;
8,3,2;
13,5,2,1;
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MAPLE
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with(combinat): T:=proc(n, k) if n=2*k+1 then 2 elif n<2*k then 0 else fibonacci(n-2*k+1) fi end: for n from 0 to 18 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000045, A000071.
Sequence in context: A008678 A058741 A074945 this_sequence A030454 A103360 A104469
Adjacent sequences: A129708 A129709 A129710 this_sequence A129712 A129713 A129714
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 12 2007
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