A000288 Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) with a(0) = a(1) = a(2) = a(3) = 1. (Formerly M3307 N1332)

1, 1, 1, 1, 4, 7, 13, 25, 49, 94, 181, 349, 673, 1297, 2500, 4819, 9289, 17905, 34513, 66526, 128233, 247177, 476449, 918385, 1770244, 3412255, 6577333, 12678217, 24438049, 47105854, 90799453, 175021573, 337364929, 650291809, 1253477764

Offset

0,5

Comments

The "standard" Tetranacci numbers with initial terms (0,0,0,1) are listed in A000078. - M. F. Hasler, Apr 20 2018

For n>=0: a(n+2) is the number of length-n words with letters {0,1,2,3} where the letter x is followed by at least x zeros, see Fxtbook link. [Joerg Arndt, Apr 08 2011]

Satisfies Benford's law [see A186191]. - N. J. A. Sloane, Feb 09 2017

References

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

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..3503 (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.

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.

W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.

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).

Index entries for sequences related to Benford's law

Formula

[a(n), a(n+1), a(n+2), a(n+3)]' = (M^n)*[1 1 1 1]', where M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 1 1 1]. E.g. [7 13 25 49]' = (M^5)*[1 1 1 1]' = [a(5), a(6), a(7), a(8)]'. Here the prime denotes transpose. - Gary W. Adamson, Feb 22 2004.

a(0) = a(1) = a(2) = a(3) = 1, a(4) = 4, a(n) = 2*a(n-1) - a(n-5). - Vincenzo Librandi, Dec 21 2010

a(n) = -2*A000078(n)-A000078(n+1)+A000078(n+3). - R. J. Mathar, Apr 07 2011

G.f.: (1 - x^2 - 2*x^3) / (1 - x - x^2 - x^3 - x^4) = 1 / (1 - x / (1 - 3*x^3 / (1 - x^2 / (1 + x / (1 - x))))). - Michael Somos, May 12 2012

G.f. A(x) = 1 + x / (1 - x / (1 - 3 * x^2 / (1 + 2 * x^2))). - Michael Somos, Jan 04 2013

Example

G.f. = 1 + x + x^2 + x^3 + 4*x^4 + 7*x^5 + 13*x^6 + 25*x^7 + 49*x^8 + ...

Maple

A000288:=(-1+z**2+2*z**3)/(-1+z**2+z**3+z+z**4); # Simon Plouffe in his 1992 dissertation

Mathematica

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 34}] (* Robert G. Wilson v, Oct 27 2005 *)
LinearRecurrence[{1, 1, 1, 1}, {1, 1, 1, 1}, 30] (* Harvey P. Dale, May 23 2011 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ x (-2 - x + x^3) / (1 + x + x^2 + x^3 - x^4), {x, 0, -n}], SeriesCoefficient[ (1 - x^2 - 2 x^3) / (1 - x - x^2 - x^3 - x^4), {x, 0, n}]]; (* Michael Somos, Aug 15 2015 *)

Prog

(Maxima) A000288[0]:1$ A000288[1]:1$ A000288[2]:1$ A000288[3]:1$ A000288[n]:=A000288[n-1] + A000288[n-2]+ A000288[n-3] + A000288[n-4]$ makelist(A000288[n], n, 0, 30); /* Martin Ettl, Oct 25 2012 */
(PARI) {a(n) = if( n<0, n = -n; polcoeff( x*(-2 - x + x^3) / (1 + x + x^2 + x^3 - x^4) + x*O(x^n), n), polcoeff( (1 - x^2 - 2*x^3) / (1 - x - x^2 - x^3 - x^4) + x*O(x^n), n))}; /* Michael Somos, Jan 04 2013 */

Crossrefs

Cf. A060455.

Cf. A000078: Tetranacci numbers with a(0) = a(1) = a(2) = 0, a(3) = 1.

Sequence in context: A341843 A248098 A229439 * A074863 A118334 A205538

Adjacent sequences: A000285 A000286 A000287 * A000289 A000290 A000291

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Robert G. Wilson v, Oct 27 2005

Status

approved