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%I M1763 #187 Mar 31 2024 17:24:52
%S 1,2,7,24,82,280,956,3264,11144,38048,129904,443520,1514272,5170048,
%T 17651648,60266496,205762688,702517760,2398545664,8189147136,
%U 27959497216,95459694592,325919783936,1112759746560,3799199418368
%N a(n) = 4*a(n-1) - 2*a(n-2) (n >= 3).
%C Gives the number of L-convex polyominoes with n cells, that is convex polyominoes where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientation of the L). - Simone Rinaldi (rinaldi(AT)unisi.it), Feb 19 2007
%C Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 2) is "size of raises in pot-limit poker, one blind, maximum raising".
%C Dimensions of the graded components of the Hopf algebra of noncommutative multi-symmetric functions of level 2. For level r, the sequence would be the INVERT transform of binomial(n+r-1,n). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
%C The sum of the numbers in the n-th row of the summatory Pascal triangle (A059576). - _Ron R. King_, Jan 22 2009
%C (1 + 2x + 7x^2 + 24x^3 + ...) = 1 / (1 - 2x - 3x^2 - 4x^3 - ...). - _Gary W. Adamson_, Jul 27 2009
%C Let M be a triangle with the odd-indexed Fibonacci numbers (1, 2, 5, 13, ...) in every column, with the leftmost column shifted upwards one row. A003480 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. The analogous operation using the even-indexed Fibonacci numbers generates A001835 starting with offset 1. - _Gary W. Adamson_, Jul 27 2010
%C a(n) is the number of generalized compositions of n when there are i+1 different types of the part i, (i=1,2,...). - _Milan Janjic_, Sep 24 2010
%C Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t + 3*t^2 + 4*t^3 + ...)),
%C an o.g.f. for A003480, then
%C A001003(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. - _Tom Copeland_, Sep 06 2011
%C Excluding the initial 1, a(n) is the 2nd subdiagonal of A228405. - _Richard R. Forberg_, Sep 02 2013
%D G. Castiglione and A. Restivo, L-convex polyominoes: a survey, Chapter 2 of K. G. Subranian et al., eds., Formal Models, Languages and Applications, World Scientific, 2015.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A003480/b003480.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from T. D. Noe)
%H D. Battaglino, J. M. Fedou, S. Rinaldi and S. Socci, <a href="https://doi.org/10.46298/dmtcs.2370">The number of k-parallelogram polyominoes</a>, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154.
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/1707.07798">(an + b)-color compositions</a>, arXiv:1707.07798 [math.CO], 2017.
%H Adrien Boussicault, Simone Rinaldi, and Samanta Socci, <a href="https://arxiv.org/abs/1501.00872">The number of directed k-convex polyominoes</a>, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See t_n.
%H Steve Butler, Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo, <a href="https://arxiv.org/abs/1702.05808">Enumerating multiplex juggling patterns</a>, arXiv:1702.05808 [math.CO], 2017.
%H P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(89)90081-2">Some sequences of integers</a>, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%H G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European J. Combin. 28 (2007), no. 6, 1724-1741.
%H Yumin Cho, Jaehyun Kim, Jang Soo Kim, and Nakyung Lee, <a href="https://arxiv.org/abs/2402.09903">Enumeration of multiplex juggling card sequences using generalized q-derivatives</a>, arXiv:2402.09903 [math.CO], 2024. See p. 6.
%H Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
%H E. Duchi, S. Rinaldi and G. Schaeffer, <a href="https://arxiv.org/abs/math/0602124">The number of Z-convex polyominoes</a>, arXiv:math/0602124 [math.CO], 2006.
%H A. Frosini and S. Rinaldi, <a href="http://puma.dimai.unifi.it/17_1_2/frosini.pdf">An object grammar for the class of L-convex polyominoes</a>, PU.M.A. Vol. 17 (2006), No. 1-2, pp. 97-110.
%H Y-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Guo/guo4.html">Some n-Color Compositions</a>, J. Int. Seq. 15 (2012) 12.1.2, eq (12).
%H Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja, Frances Y. Kuo, and Ian H. Sloan, <a href="https://arxiv.org/abs/2210.17329">Uncertainty quantification for random domains using periodic random variables</a>, arXiv:2210.17329 [math.NA], 2022.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=418">Encyclopedia of Combinatorial Structures 418</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions</a>, arXiv:0806.3682 [math.CO], 2008.
%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 J. Riordan, <a href="https://doi.org/10.1090/S0025-5718-1975-0366686-9">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.
%H J. Riordan, <a href="/A003480/a003480.pdf">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
%H <a href="/index/Poi#poker">Index entries for sequences related to poker</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F a(n) = (n+1)*a(0) + n*a(1) + ... + 3*a(n-2) + 2*a(n-1). - _Amarnath Murthy_, Aug 17 2002
%F G.f.: (1-x)^2/(1-4*x+2*x^2). - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = A007070(n)/2, n > 0.
%F G.f.: 1/( 1 - Sum_{k>=1} (k+1)*x^k ).
%F a(n+1)*a(n+1) - a(n+2)*a(n) = 2^n, n > 0. - D. G. Rogers, Jul 12 2004
%F For n > 0, a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/(4*sqrt(2)). - _Rolf Pleisch_, Aug 03 2009
%F If the leading 1 is removed, 2, 7, 24, ... is the binomial transform of 2, 5, 12, 29, ..., which is A000129 without its first 2 terms, and the second binomial transform of 2, 3, 4, 6, ..., which is A029744, again without its leading 1. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
%F a(n) = Sum((1+p_1)(1+p_2)...)(1+p_m)), summation being over all compositions (p_1, p_2, ..., p_m) of n. Example: a(3)=24; indeed, the compositions of 3 are (1,1,1), (1,2),(2,1), (3) and we have 2*2*2 + 2*3 + 3*2 + 4 = 24. - _Emeric Deutsch_, Oct 17 2010
%F a(n) = Sum_{k>=0} binomial(n+2*k-1,n) / 2^(k+1). - _Vaclav Kotesovec_, Dec 31 2013
%p INVERT([seq(n+1,n=1..20)]); # Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
%t a[0]=1; a[1]=2; a[2]=7; a[n_]:=a[n]=4*a[n-1] - 2*a[n-2]; Table[a[n],{n,0,24}] (* _Jean-François Alcover_, Mar 22 2011 *)
%t Join[{1},LinearRecurrence[{4,-2},{2,7},40]] (* _Harvey P. Dale_, Oct 23 2011 *)
%o (PARI) a(n)=polcoeff((1-x)^2/(1-4*x+2*x^2)+x*O(x^n),n)
%o (PARI) a(n)=local(x); if(n<1,n==0,x=(2+quadgen(8))^n; imag(x)+real(x)/2)
%o (Haskell)
%o a003480 n = a003480_list !! n
%o a003480_list = 1 : 2 : 7 : (tail $ zipWith (-)
%o (tail $ map (* 4) a003480_list) (map (* 2) a003480_list))
%o -- _Reinhard Zumkeller_, Jan 16 2012, Oct 03 2011
%Y Row sums of A059576 and of A181289. Second differences of A007070.
%Y Cf. A007052, A126764.
%Y Cf. A001835, A006012, A145839, A145840, A145841.
%Y Column k=2 of A261780.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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