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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, A122189.
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
| 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
| 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
| O.g.f.: 1/(1-x-x^2-x^3-x^4-x^5-x^6-x^7); 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
For a(0)=a(1)=1, a(2)=2, a(3)=4, a(4)=8, a(5)=16, a(6)=32, a(7)=64, a(n)=2*a(n-1)-a(n-8). [From Vincenzo Librandi, Dec 20 2010]
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MATHEMATICA
| a[0] = a[1] = 1; a[2] = 2; a[3] = 4; a[4] = 8; a[5] = 16; a[6] = 32; a[7] = 64; a[n_] := 2*a[n - 1] - a[n - 8]; Array[a, 31, 0]
CoefficientList[ Series[(1 - x)/(1 - 2 x + x^8), {x, 0, 30}], x]
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CROSSREFS
| 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: A062257 A172316 A062258 * A122189 A194630 A133024
Adjacent sequences: A066175 A066176 A066177 * A066179 A066180 A066181
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KEYWORD
| nonn
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AUTHOR
| Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 14 2001
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EXTENSIONS
| Definition corrected by Vincenzo Librandi, Dec 20 2010
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