%I #68 Jan 27 2023 13:42:01
%S 1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,
%T 465,616,816,1081,1432,1897,2513,3329,4410,5842,7739,10252,13581,
%U 17991,23833,31572,41824,55405,73396,97229,128801,170625,226030,299426,396655,525456,696081,922111,1221537,1618192,2143648,2839729,3761840,4983377,6601569,8745217
%N Expansion of 1/(1-x^2-x^3).
%C Number of compositions (ordered partitions) into parts 2 and 3. - _Joerg Arndt_, Aug 21 2013
%C a(n) is the top left entry of the n-th power of any of the 3X3 matrices [0, 1, 1; 0, 0, 1; 1, 0, 0], [0, 1, 0; 1, 0, 1; 1, 0, 0], [0, 1, 1; 1, 0, 0; 0, 1, 0] or [0, 0, 1; 1, 0, 0; 1, 1, 0]. - _R. J. Mathar_, Feb 03 2014
%C Conjectured values of d(n), the dimension of a Z-module in MZV(conv). See the Waldschmidt link. - _Michael Somos_, Mar 14 2014
%C Shannon et al. (2006) call these the Van der Laan numbers. - _N. J. A. Sloane_, Jan 11 2022
%D A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See R_n.
%D Michel Waldschmidt, "Multiple Zeta values and Euler-Zagier numbers", in Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).
%H Vincenzo Librandi, <a href="/A182097/b182097.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Hoffman, <a href="http://dx.doi.org/10.1006/jabr.1997.7127">The algebra of multiharmonic series</a>, Journ. of Alg., Vol. 192, Issue 2 (Aug 1997), 477-495.
%H I. E. Leonard and A. C. F. Liu, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.119.04.333">A familiar recurrence occurs again</a>, Amer. Math. Monthly, 119 (2012), 333-336.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1406.7788">Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices</a>, arXiv:1406.7788 (2014), eq. (32).
%H Michel Waldschmidt, <a href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Transparents.pdf">Multiple Zeta values and Euler-Zagier numbers</a>, Slides, Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F G.f.: 1 / (1 - x^2 - x^3).
%F a(n) = A000931(n+3).
%F From _Michael Somos_, Dec 13 2013: (Start)
%F a(n) = A176971(-n).
%F a(n) = a(n-2) + a(n-3) for all n in Z.
%F a(n-7) = A133034(n).
%F a(n-5) = A078027(n).
%F a(n-3) = A000931(n).
%F a(n+2) = A134816(n).
%F a(n+4) = A164001(n) if n > 1. - (End)
%F a(n) = (A001608(n) - A000931(n))/2. - _Elmo R. Oliveira_, Dec 31 2022
%e G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
%t a[ n_] := If[n < 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, n}]]; (* _Michael Somos_, Dec 13 2013 *)
%t CoefficientList[Series[1/(1-x^2-x^3),{x,0,60}],x] (* or *) LinearRecurrence[ {0,1,1},{1,0,1},70] (* _Harvey P. Dale_, Dec 04 2014 *)
%o (PARI) {a(n) = if( n<0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n))}; /* _Michael Somos_, Dec 13 2013 */
%o (PARI) Vec(1/(1-x^2-x^3) + O(x^99)) \\ _Altug Alkan_, Sep 02 2016
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3))); // _G. C. Greubel_, Aug 11 2018
%Y The following are basically all variants of the same sequence: A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.
%Y Cf. A000931, A001608.
%K nonn,easy
%O 0,6
%A _N. J. A. Sloane_, Apr 11 2012
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