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A055099 Expansion of g.f.: (1 + x)/(1 - 3*x - 2*x^2).

%I A055099 #109 May 24 2024 22:09:50
%S A055099 1,4,14,50,178,634,2258,8042,28642,102010,363314,1293962,4608514,
%T A055099 16413466,58457426,208199210,741512482,2640935866,9405832562,
%U A055099 33499369418,119309773378,424928058970,1513403723666,5390067288938,19197009314146,68371162520314,243507506189234
%N A055099 Expansion of g.f.: (1 + x)/(1 - 3*x - 2*x^2).
%C A055099 Row sums of triangle A054458.
%C A055099 a(n) = term (1,1) in M^n, M = the 3 X 3 matrix [1,1,1; 1,1,1; 2,2,1]. - _Gary W. Adamson_, Mar 12 2009
%C A055099 Equals the INVERT transform of A001333: (1, 3, 7, 17, 41, 99, ...). - _Gary W. Adamson_, Aug 14 2010
%C A055099 a(n) is the number of one sided n-step walks taking steps from {(0,1), (-1,0), (1,0), (1,1)}. - _Shanzhen Gao_, May 13 2011
%C A055099 Number of quaternary words of length n on {0,1,2,3} containing no subwords 03 or 30. - _Philippe Deléham_, Apr 27 2012
%C A055099 Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 2, 272, 12, 18, 24, ... - _R. J. Mathar_, Aug 10 2012
%C A055099 a(n) = A007481(2*n+1) - A007481(2*n) = A007481(2*(n+1)) - A007481(2*n+1). - _Reinhard Zumkeller_, Oct 25 2015
%C A055099 Number of length-n words on a,b,c,d avoiding aa and ab. For n >= 1, the number of such words ending with a or the number of those ending with b is A007482(n-1), and the number of those ending with c or the number of those ending with d is a(n-1). - _Jianing Song_, Jun 01 2022
%D A055099 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.4.6).
%H A055099 Alois P. Heinz, <a href="/A055099/b055099.txt">Table of n, a(n) for n = 0..1000</a>
%H A055099 M. Abrate, S. Barbero, U. Cerruti, and N. Murru, <a href="http://dx.doi.org/10.1155/2013/543913">Construction and composition of rooted trees via descent functions</a>, Algebra, Volume 2013 (2013), Article ID 543913, 11 pages.
%H A055099 D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3 , example 17
%H A055099 A. S. Fraenkel, <a href="https://arxiv.org/abs/math/9809074">Heap games, numeration systems and sequences</a>, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
%H A055099 Shanzhen Gao and Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.
%H A055099 S. Gao and H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, 2010.
%H A055099 Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1304.4286">(a,b)-rectangle patterns in permutations and words</a>, arXiv:1304.4286 [math.CO], 2013.
%H A055099 Paul K. Stockmeyer, <a href="http://arxiv.org/abs/1504.04404">The Pascal Rhombus and the Stealth Configuration</a>, arXiv:1504.04404 [math.CO], 2015.
%H A055099 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,2).
%F A055099 a(n) = a*c^n - b*d^n, a := (5 + sqrt(17))/(2*sqrt(17)), b := (5 - sqrt(17))/(2*sqrt(17)), c := (3 + sqrt(17))/2, d := (3 - sqrt(17))/2.
%F A055099 a(n) = Sum_{m=0..n} A054458(n, m).
%F A055099 a(n) = F32(n) + F32(n-1) with F32(n) = A007482(n), n >= 1, a(0) = 1.
%F A055099 a(n) = A007482(n) + A007482(n-1) = 2*A007482(n) - A104934(n). - _R. J. Mathar_, Jul 23 2010
%F A055099 a(n) = 3*a(n-1) + 2*a(n-2) with a(0) = 1, a(1) = 4. - _Vincenzo Librandi_, Dec 08 2010
%F A055099 a(n) = (Sum_{k = 0..n} A202396(n,k)*3^k)/2^n. - _Philippe Deléham_, Feb 05 2012
%F A055099 a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - _G. C. Greubel_, Jun 27 2021
%F A055099 a(n) = 2*a(n-1) + 2*A007482(n-1), n >= 1. - _Jianing Song_, Jun 01 2022
%F A055099 E.g.f.: exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2))/17. - _Stefano Spezia_, May 24 2024
%e A055099 a(3) = 50 because among the 4^3 = 64 quaternary words of length 3 only 14 namely 003, 030, 031, 032, 033, 103, 130, 203, 230, 300, 301, 302, 303, 330 contain the subwords 03 or 30. - _Philippe Deléham_, Apr 27 2012
%p A055099 a := proc(n) option remember; `if`(n < 2, [1, 4][n+1], (3*a(n-1) + 2*a(n-2))) end:
%p A055099 seq(a(n), n=0..23); # _Peter Luschny_, Jan 06 2019
%t A055099 max = 24; cv = ContinuedFraction[ Sqrt[2], max] // Convergents // Numerator; Series[ 1/(1 - cv.x^Range[max]), {x, 0, max}] // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Jun 21 2013, after _Gary W. Adamson_ *)
%t A055099 LinearRecurrence[{3, 2}, {1, 4}, 24] (* _Jean-François Alcover_, Sep 23 2017 *)
%o A055099 (Haskell)
%o A055099 a055099 n = a007481 (2 * n + 1) - a007481 (2 * n)
%o A055099 -- _Reinhard Zumkeller_, Oct 25 2015
%o A055099 (Magma) I:=[1,4]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Jun 27 2021
%o A055099 (Sage) [(i*sqrt(2))^(n-1)*( i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # _G. C. Greubel_, Jun 27 2021
%Y A055099 Column k=2 of A255256.
%Y A055099 Cf. A001333, A002203, A007481, A007482, A054458, A104934, A202396.
%K A055099 easy,nonn
%O A055099 0,2
%A A055099 _Wolfdieter Lang_, Apr 26 2000
%E A055099 Edited by _N. J. A. Sloane_, Jun 08 2010