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Search: id:A066178
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| A066178 |
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Number of binary bit strings of length n with no block of 8 or more 0's. Nonzero heptanacci numbers (cf. A066178). |
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+0
17
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| 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Analogous bit string description and O.g.f. (1-x)/(1-2x+x^{k+1}) works for nonzero k-nacci numbers.
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REFERENCES
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Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Heptanacci Number
Du, Zhao Hui, Link giving derivation and proof of the formula
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FORMULA
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O.g.f.: (1-x)/(1-2x+x^8); a(n)=sum(a(i), i=n-7..n-1).
a(n)=round({r-1}/{(t+1)r-2t} * r^{n-1}), where r is the heptanacci constant, the real root of the equation x^{t+1)-2x^t+1=0 which is greater than 1. The formula could also be used for a k-step Fibonacci sequence if r is replaced by the k-bonacci constant, as in A000045, A000073, A000078, A001591, A001592 - Du, Zhao Hui (zhao.hui.du(AT)gmail.com), Aug 24 2008
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CROSSREFS
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Cf. A000045 (k=2, Fibonacci numbers), A000073 (k=3, tribonacci) A000078 (k=4, tetranacci) A001591 (k=5, pentanacci) A001592 (k=6, hexanacci), A122189 (k=7, heptanacci).
Row 7 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Sequence in context: A145113 A062257 A062258 this_sequence A122189 A133024 A060376
Adjacent sequences: A066175 A066176 A066177 this_sequence A066179 A066180 A066181
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KEYWORD
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nonn
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AUTHOR
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Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 14 2001
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