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.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Cf. A000045, A000288, A000383, A060455, A186192.

Cf. A001591 (Pentanacci numbers starting 0, 0, 0, 0, 1).

Programs

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] ];

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]

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