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A000322 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)
49
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

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.

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 and Conjectures, Université du Québec à Montréal, 1992.

MAPLE

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

MATHEMATICA

LinearRecurrence[{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, 50]

PROG

(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

CROSSREFS

Cf. A000045, A000288, A000383, A060455.

Sequence in context: A160426 A258411 A059743 * A205539 A020737 A147401

Adjacent sequences:  A000319 A000320 A000321 * A000323 A000324 A000325

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 2 04:03 EDT 2015. Contains 259128 sequences.