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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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26
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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
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OFFSET
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0,3
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COMMENTS
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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.
Compositions of n into parts <= 7. - Joerg Arndt, Aug 06 2012
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Zhao Hui Du, Link giving derivation and proof of the formula
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Heptanacci Number
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1, 1, 1).
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FORMULA
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O.g.f.: 1/(1-x-x^2-x^3-x^4-x^5-x^6-x^7).
a(n) = Sum_{i=n-7..n-1} a(i).
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. - Zhao Hui Du, Aug 24 2008
a(n) = 2*a(n-1) - a(n-8). - Vincenzo Librandi, Dec 20 2010
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MATHEMATICA
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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]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {1, 1, 2, 4, 8, 16, 32}, 40] (* Harvey P. Dale, Nov 16 2014 *)
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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: A172316 A062258 A239560 * A122189 A194630 A251672
Adjacent sequences: A066175 A066176 A066177 * A066179 A066180 A066181
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KEYWORD
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nonn,easy
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AUTHOR
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Len Smiley, Dec 14 2001
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EXTENSIONS
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Definition corrected by Vincenzo Librandi, Dec 20 2010
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STATUS
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approved
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