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Search: id:A129712
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| A129712 |
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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 10'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, 4, 1, 6, 1, 1, 10, 2, 1, 16, 3, 1, 1, 26, 5, 2, 1, 42, 8, 3, 1, 1, 68, 13, 5, 2, 1, 110, 21, 8, 3, 1, 1, 178, 34, 13, 5, 2, 1, 288, 55, 21, 8, 3, 1, 1, 466, 89, 34, 13, 5, 2, 1, 754, 144, 55, 21, 8, 3, 1, 1, 1220, 233, 89, 34, 13, 5, 2, 1, 1974, 377, 144, 55, 21, 8, 3, 1, 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)=A052952(n-2) (n>=2).
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FORMULA
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T(0,0)=1, T(n,0)=2F(n) for n>=1, T(2k,k)=T(2k+1,k)=1 for k>=1, T(n,k)=F(n-2k) for 1<=k<(n-1)/2. G.f.=G(t,z)=(1+z-z^2-t*z^3)/[(1-z-z^2)(1-t*z^2)].
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EXAMPLE
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T(7,2)=2 because we have 1010110 and 1010111.
Triangle starts:
1;
2;
2,1;
4,1;
6,1,1;
10,2,1;
16,3,1,1;
26,5,2,1;
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MAPLE
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with(combinat): T:=proc(n, k) if k=0 and n=0 then 1 elif k=0 then 2*fibonacci(n) elif n=2*k or n=2*k+1 then 1 elif n>2*k+1 then fibonacci(n-2*k) else 0 fi end: for n from 0 to 18 do seq(T(n, k), k=0..floor(n/2)) od;
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CROSSREFS
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Cf. A000045, A052952.
Sequence in context: A018219 A116633 A134666 this_sequence A051720 A022693 A146478
Adjacent sequences: A129709 A129710 A129711 this_sequence A129713 A129714 A129715
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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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