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%I M3307 N1332 #107 Jan 16 2024 04:45:05
%S 1,1,1,1,4,7,13,25,49,94,181,349,673,1297,2500,4819,9289,17905,34513,
%T 66526,128233,247177,476449,918385,1770244,3412255,6577333,12678217,
%U 24438049,47105854,90799453,175021573,337364929,650291809,1253477764
%N 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.
%C The "standard" Tetranacci numbers with initial terms (0,0,0,1) are listed in A000078. - _M. F. Hasler_, Apr 20 2018
%C 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]
%C Satisfies Benford's law [see A186191]. - _N. J. A. Sloane_, Feb 09 2017
%D Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Indranil Ghosh, <a href="/A000288/b000288.txt">Table of n, a(n) for n = 0..3503</a> (terms 0..200 from T. D. Noe)
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp.311-312.
%H B. G. Baumgart, Letter to the editor, <a href="http://www.fq.math.ca/Scanned/2-4/baumgart-a.pdf">Part 1</a>, <a href="http://www.fq.math.ca/Scanned/2-4/baumgart-b.pdf">Part 2</a>, <a href="http://www.fq.math.ca/Scanned/2-4/baumgart-c.pdf">Part 3</a>, Fib. Quart. 2 (1964), 260, 302.
%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H W. C. Lynch, <a href="http://www.fq.math.ca/Scanned/8-1/lynch.pdf">The t-Fibonacci numbers and polyphase sorting</a>, Fib. Quart., 8 (1970), pp. 6ff.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Álvaro Serrano Holgado and Luis Manuel Navas Vicente, <a href="https://arxiv.org/abs/2301.11747">The zeta function of a recurrence sequence of arbitrary degree</a>, arXiv:2301.11747 [math.NT], 2023.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1).
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F [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.
%F 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
%F a(n) = -2*A000078(n)-A000078(n+1)+A000078(n+3). - _R. J. Mathar_, Apr 07 2011
%F 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
%F G.f. A(x) = 1 + x / (1 - x / (1 - 3 * x^2 / (1 + 2 * x^2))). - _Michael Somos_, Jan 04 2013
%e G.f. = 1 + x + x^2 + x^3 + 4*x^4 + 7*x^5 + 13*x^6 + 25*x^7 + 49*x^8 + ...
%p A000288:=(-1+z**2+2*z**3)/(-1+z**2+z**3+z+z**4); # _Simon Plouffe_ in his 1992 dissertation
%t 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 *)
%t LinearRecurrence[{1,1,1,1},{1,1,1,1},30] (* _Harvey P. Dale_, May 23 2011 *)
%t 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 *)
%o (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 */
%o (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 */
%Y Cf. A060455.
%Y Cf. A000078: Tetranacci numbers with a(0) = a(1) = a(2) = 0, a(3) = 1.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Oct 27 2005
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