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 A003480 a(n) = 4a(n-1) - 2a(n-2) (n >= 3). (Formerly M1763) 43
 1, 2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048, 17651648, 60266496, 205762688, 702517760, 2398545664, 8189147136, 27959497216, 95459694592, 325919783936, 1112759746560, 3799199418368 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 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". 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 The sum of the numbers in the n-th row of the summatory Pascal triangle (A059576). - Ron R. King, Jan 22 2009 (1 + 2x + 7x^2 + 24x^3 + ...) = 1 / (1 - 2x - 3x^2 - 4x^3 - ...). - Gary W. Adamson, Jul 27 2009 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->inf} 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 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 Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t + 3*t^2 + 4*t^3 + ...)),   an o.g.f. for A003480, then   A001003(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. - Tom Copeland, Sep 06 2011 Excluding initial 1, a(n) is the 2nd sub-diagonal of A228405. - Richard R. Forberg, Sep 02 2013 REFERENCES 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. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe) D. Battaglino, J. M. Fedou, S. Rinaldi and S. Socci, The number of k-parallelogram polyominoes, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154. Daniel Birmajer, Juan B. Gil, Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017. Adrien Boussicault, Simone Rinaldi, and Samanta Socci, The number of directed k-convex polyominoes, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See t_n. Steve Butler, Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo, Enumerating multiplex juggling patterns, arXiv:1702.05808 [math.CO], 2017. P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102. G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741. Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. E. Duchi, S. Rinaldi and G. Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006. A. Frosini and S. Rinaldi, An object grammar for the class of L-convex polyominoes, PU.M.A. Vol. 17 (2006), No. 1-2, pp. 97-110. Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (12). INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 418 M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7. J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy] Index entries for linear recurrences with constant coefficients, signature (4,-2). FORMULA a(n) = (n+1)*a(0) + n*a(1) + ... + 3*a(n-2) + 2*a(n-1). - Amarnath Murthy, Aug 17 2002 G.f.: (1-x)^2/(1-4*x+2*x^2). - Simon Plouffe in his 1992 dissertation a(n) = A007070(n)/2, n > 0. G.f.: 1/( 1 - Sum_{k>=1} (k+1)*x^k ). a(n+1)*a(n+1) - a(n+2)*a(n) = 2^n, n > 0. - D. G. Rogers, Jul 12 2004 For n > 0, a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/(4*sqrt(2)). - Rolf Pleisch, Aug 03 2009 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 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 a(n) = Sum_{k>=0} binomial(n+2*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013 MAPLE INVERT([seq(n+1, n=1..20)]); # Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008 MATHEMATICA a=1; a=2; a=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 *) Join[{1}, LinearRecurrence[{4, -2}, {2, 7}, 40]] (* Harvey P. Dale, Oct 23 2011 *) PROG (PARI) a(n)=polcoeff((1-x)^2/(1-4*x+2*x^2)+x*O(x^n), n) (PARI) a(n)=local(x); if(n<1, n==0, x=(2+quadgen(8))^n; imag(x)+real(x)/2) (Haskell) a003480 n = a003480_list !! n a003480_list = 1 : 2 : 7 : (tail \$ zipWith (-)    (tail \$ map (* 4) a003480_list) (map (* 2) a003480_list)) -- Reinhard Zumkeller, Jan 16 2012, Oct 03 2011 CROSSREFS Row sums of A059576 and of A181289. Second differences of A007070. Cf. A007052, A126764. Cf. A001835, A006012, A145839, A145840, A145841. Column k=2 of A261780. Sequence in context: A027128 A099463 A021000 * A020727 A329274 A088854 Adjacent sequences:  A003477 A003478 A003479 * A003481 A003482 A003483 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified July 28 23:26 EDT 2021. Contains 346340 sequences. (Running on oeis4.)