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A119458
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Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 00 (0<=k<=n).
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3
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1, 1, 1, 3, 0, 1, 4, 3, 0, 1, 7, 4, 4, 0, 1, 11, 10, 5, 5, 0, 1, 18, 18, 15, 6, 6, 0, 1, 29, 35, 28, 21, 7, 7, 0, 1, 47, 64, 60, 40, 28, 8, 8, 0, 1, 76, 117, 117, 93, 54, 36, 9, 9, 0, 1, 123, 210, 230, 190, 135, 70, 45, 10, 10, 0, 1, 199, 374, 440, 396, 286, 187, 88, 55, 11, 11, 0, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Sum of entries in row n is 2^n (A000079). T(n,0)=A000204(n) for n>=1 (Lucas numbers). T(n,1)=A006490(n). T(n,2)=A006491(n-1). Sum(k*T(n,k),k=0..n)=A057711(n). In Carlitz and Scoville (p. 252) the first term is 2
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REFERENCES
| L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
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FORMULA
| T(n,k)=T(n-1,k)+T(n-2,k)+T(n-1,k-1)-T(n-2,k-1) for n>=3 and k>=1. G.f.=G(t,z)=(1+z^2-tz^2)/(1-z-z^2-tz+tz^2).
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EXAMPLE
| T(5,3)=5 because we have 10000,01000,00100,00010 and 00001.
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MAPLE
| T:=proc(n, k): if k>n or k<0 then 0 elif n=0 and k=0 then 1 elif n=1 then 1 elif n=2 and k=0 then 3 elif n=2 and k=1 then 0 else T(n-1, k)+T(n-2, k)+T(n-1, k-1)-T(n-2, k-1) fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000079, A000204, A006490, A006491, A057711.
Sequence in context: A177330 A197126 A048963 * A106356 A091613 A039727
Adjacent sequences: A119455 A119456 A119457 * A119459 A119460 A119461
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 20 2006
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