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Search: id:A129710
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| A129710 |
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Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (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, 2, 3, 2, 5, 1, 2, 7, 4, 2, 9, 9, 1, 2, 11, 16, 5, 2, 13, 25, 14, 1, 2, 15, 36, 30, 6, 2, 17, 49, 55, 20, 1, 2, 19, 64, 91, 50, 7, 2, 21, 81, 140, 105, 27, 1, 2, 23, 100, 204, 196, 77, 8, 2, 25, 121, 285, 336, 182, 35, 1, 2, 27, 144, 385, 540, 378, 112, 9, 2, 29, 169, 506
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Also number of Fibonacci binary words of length n and having k 10 subwords. Row n has 1+floor(n/2) terms. Row sums are the Fibonacci numbers (A000045). T(n,0)=2 for n>=1. Sum(k*T(n,k), k>=0)=A023610(n-2).
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FORMULA
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T(n,k)=binom(n-k,k)+binom(n-k-1,k) for n>=1 and 0<=k<=floor(n/2). G.f.=G(t,z)=(1+z)/(1-z-tz^2).
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EXAMPLE
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T(5,2)=4 because we have 10101, 01101, 01010 and 01011.
Triangle starts:
1;
2;
2,1;
2,3;
2,5,1;
2,7,4;
2,9,9,1;
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MAPLE
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T:=proc(n, k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k, k)+binomial(n-k-1, k) else 0 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, A023610.
Sequence in context: A029251 A133091 A112204 this_sequence A070680 A054711 A134658
Adjacent sequences: A129707 A129708 A129709 this_sequence A129711 A129712 A129713
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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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