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)

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

Offset

0,6

Comments

For n>=0: a(n+2) is the number of length-n words 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

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

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, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.

Martin Burtscher, Igor Szczyrba, and 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; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Álvaro Serrano Holgado and Luis Manuel Navas Vicente, The zeta function of a recurrence sequence of arbitrary degree, arXiv:2301.11747 [math.NT], 2023.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Index entries for sequences related to Benford's law

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, 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

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved