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A000322
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Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.
(Formerly M3786 N1542)
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53
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1, 1, 1, 1, 1, 5, 9, 17, 33, 65, 129, 253, 497, 977, 1921, 3777, 7425, 14597, 28697, 56417, 110913, 218049, 428673, 842749, 1656801, 3257185, 6403457, 12588865, 24749057, 48655365, 95653929, 188050673, 369697889, 726806913, 1428864769
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OFFSET
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0,6
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COMMENTS
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For n>=0: a(n+2) is the number of length-n strings with letters {0,1,2,3,4} where the letter x is followed by at least x zeros, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Satisfies Benford's law [see A186192] - N. J. A. Sloane, Feb 09 2017
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Indranil Ghosh, Table of n, a(n) for n = 0..3402 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), pp.311-312.
B. G. Baumgart, Letter to the editor Part 1 Part 2 Part 3, Fib. Quart. 2 (1964), 260, 302.
D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet , J. Int. Seq. 19 (2016) # 16.1.3, Example 7
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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1).
Index entries for sequences related to Benford's law
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MAPLE
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A000322:=(-1+z**2+2*z**3+3*z**4)/(-1+z**2+z**3+z+z**4+z**5); # Simon Plouffe in his 1992 dissertation.
a:= n-> (Matrix([[1$5]]). Matrix(5, (i, j)-> if (i=j-1) or j=1 then 1 else 0 fi)^n)[1, 5]: seq (a(n), n=0..28); # Alois P. Heinz, Aug 26 2008
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MATHEMATICA
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LinearRecurrence[{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, 50]
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PROG
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(MAGMA) [ n le 5 select 1 else Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4)+Self(n-5): n in [1..40] ];
(PARI) Vec((1-x^2-2*x^3-3*x^4)/(1-x-x^2-x^3-x^4-x^5)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2013
(J) (see www.jsoftware.com) First construct the generating matrix
(((+ +/), ]), :^:(1=#@$))/&.|.<:/~i.5
1 1 1 1 1
1 2 2 2 2
2 3 4 4 4
4 6 7 8 8
8 12 14 15 16
Given that matrix, one can produce the first 2000 numbers in almost 17 millisecs by
, ((((+ +/), ]), :^:(1=#@$))/&.|.<:/~i.5) (+/ . *)^:(i.400) 1 1 1 1 1x
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CROSSREFS
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Cf. A000045, A000288, A000383, A060455, A186192.
Cf. A001591 (Pentanacci numbers starting 0, 0, 0, 0, 1).
Sequence in context: A301786 A258411 A059743 * A205539 A020737 A262452
Adjacent sequences: A000319 A000320 A000321 * A000323 A000324 A000325
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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