Search: id:A066178
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%I A066178
%S A066178 1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936,15808,31489,62725,
%T A066178 124946,248888,495776,987568,1967200,3918592,7805695,15548665,30972384,
%U A066178 61695880,122895984,244804400,487641600
%N A066178 Number of binary bit strings of length n with no block of 8 or more 0's. 
               Nonzero heptanacci numbers (cf. A066178).
%C A066178 Analogous bit string description and O.g.f. (1-x)/(1-2x+x^{k+1}) works 
               for nonzero k-nacci numbers.
%D A066178 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.
%H A066178 T. D. Noe, <a href="b066178.txt">Table of n, a(n) for n=0..200</a>
%H A066178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A066178 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HeptanacciNumber.html">Heptanacci Number</a>
%H A066178 Du, Zhao Hui, <a href="http://bbs.emath.ac.cn/viewthread.php?tid=667&page=4&fromuid=20#pid9145">
               Link giving derivation and proof of the formula</a>
%F A066178 O.g.f.: (1-x)/(1-2x+x^8); a(n)=sum(a(i), i=n-7..n-1).
%F A066178 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
%Y A066178 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).
%Y A066178 Row 7 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A066178 Sequence in context: A145113 A062257 A062258 this_sequence A122189 A133024 
               A060376
%Y A066178 Adjacent sequences: A066175 A066176 A066177 this_sequence A066179 A066180 
               A066181
%K A066178 nonn
%O A066178 0,3
%A A066178 Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 14 2001

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