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Search: id:A129717
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| A129717 |
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Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 101's (n>=0, 0<=k<=floor((n-1)/2))). A Fibonacci binary word is a binary word having no 00 subword. |
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+0
2
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| 1, 2, 3, 4, 1, 4, 4, 4, 8, 1, 4, 12, 5, 4, 16, 13, 1, 4, 20, 25, 6, 4, 24, 41, 19, 1, 4, 28, 61, 44, 7, 4, 32, 85, 85, 26, 1, 4, 36, 113, 146, 70, 8, 4, 40, 145, 231, 155, 34, 1, 4, 44, 181, 344, 301, 104, 9, 4, 48, 221, 489, 532, 259, 43, 1, 4, 52, 265, 670, 876, 560, 147, 10, 4, 56
(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-1)/2) terms for n>=1. Row sums are the Fibonacci numbers (A000045). T(n,1)=A008574(n-3). T(n,2)=A001844(n-5). T(n,3)=A005900(n-6). T(n,4)=A006325(n-7). T(n,5)=A033455(n-10). T(n,k)=A129718(n,k+1) (since in each word: 1 + the number of 101's = number of runs of 1's). Sum(k*T(n,k), k>=0)=A004798(n-2).
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FORMULA
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G.f.=G(t,z)=(1+z)(1+z^2-tz^2)/(1-z-tz^2). G.f. of col 0 = (1+z)(1+z^2)/(1-z), leading to the partial sums of 1,1,1,1,0,0,0,... . G.f. of col k = z^(2k+1)*(1+z)^2/(1-z)^(k+1) (k>=1). T(n,k)=binom(n-k-1,k)+2binom(n-k-2,k)+binom(n-k-3,k) for n>=4 and 0<=k<n/2.
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EXAMPLE
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T(6,2)=5 because we have 110101, 101101, 101010, 101011 and 010101.
Triangle starts:
1;
2;
3;
4,1;
4,4;
4,8,1;
4,12,5;
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MAPLE
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T:=proc(n, k) if n=0 and k=0 then 1 elif n=1 and k=0 then 2 elif n=2 and k=0 then 3 elif n=3 and k=1 then 1 elif k<n/2 then binomial(n-k-1, k)+2*binomial(n-k-2, k)+binomial(n-k-3, k) else 0 fi end: 1; for n from 1 to 18 do seq(T(n, k), k=0..floor(n-1)/2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000045, A008574, A001844, A005900, A006325, A033455, A129718, A004798.
Sequence in context: A159798 A003324 A110630 this_sequence A117742 A117716 A097150
Adjacent sequences: A129714 A129715 A129716 this_sequence A129718 A129719 A129720
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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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