cp's OEIS Frontend

A182097 Expansion of 1/(1-x^2-x^3).

%I A182097 #74 Jan 10 2025 04:38:54
%S A182097 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 A182097 465,616,816,1081,1432,1897,2513,3329,4410,5842,7739,10252,13581,
%U A182097 17991,23833,31572,41824,55405,73396,97229,128801,170625,226030,299426,396655,525456,696081,922111,1221537,1618192,2143648,2839729,3761840,4983377,6601569,8745217
%N A182097 Expansion of 1/(1-x^2-x^3).
%C A182097 Number of compositions (ordered partitions) into parts 2 and 3. - _Joerg Arndt_, Aug 21 2013
%C A182097 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 A182097 Conjectured values of d(n), the dimension of a Z-module in MZV(conv). See the Waldschmidt link. - _Michael Somos_, Mar 14 2014
%C A182097 Shannon et al. (2006) call these the Van der Laan numbers. - _N. J. A. Sloane_, Jan 11 2022
%D A182097 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 A182097 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 A182097 Vincenzo Librandi, <a href="/A182097/b182097.txt">Table of n, a(n) for n = 0..1000</a>
%H A182097 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 A182097 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 A182097 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 A182097 Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, <a href="https://doi.org/10.9734/ajpas/2024/v26i12691">A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers</a>, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
%H A182097 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 A182097 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F A182097 G.f.: 1 / (1 - x^2 - x^3).
%F A182097 a(n) = A000931(n+3).
%F A182097 From _Michael Somos_, Dec 13 2013: (Start)
%F A182097 a(n) = A176971(-n).
%F A182097 a(n) = a(n-2) + a(n-3) for all n in Z.
%F A182097 a(n-7) = A133034(n).
%F A182097 a(n-5) = A078027(n).
%F A182097 a(n-3) = A000931(n).
%F A182097 a(n+2) = A134816(n).
%F A182097 a(n+4) = A164001(n) if n > 1. - (End)
%F A182097 a(n) = (A001608(n) - A000931(n))/2. - _Elmo R. Oliveira_, Dec 31 2022
%e A182097 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 A182097 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 A182097 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 A182097 (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 A182097 (PARI) Vec(1/(1-x^2-x^3) + O(x^99)) \\ _Altug Alkan_, Sep 02 2016
%o A182097 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3))); // _G. C. Greubel_, Aug 11 2018
%Y A182097 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 A182097 Cf. A000931, A001608.
%K A182097 nonn,easy
%O A182097 0,6
%A A182097 _N. J. A. Sloane_, Apr 11 2012