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 A055588 a(n) = 3a(n-1) - a(n-2) - 1 with a(0)=1, a(1)=2. 16
 1, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of directed column-convex polyominoes with area n+2 and having two cells in the bottom row. - Emeric Deutsch, Jun 14 2001 a(n) = length of the list generated by the substitution: 3->3, 4->(3,4,6), 6->(3,4,6,6): {3, 4}, {3, 3, 4, 6}, {3, 3, 3, 4, 6, 3, 4, 6, 6}, {3, 3, 3, 3, 4, 6, 3, 4, 6, 6, 3, 3, 4, 6, 3, 4, 6, 6, 3, 4, 6, 6}, etc. - Wouter Meeussen, Nov 23, 2003 Equals row sums of triangle A144955. - Gary W. Adamson, Sep 27 2008 Equals the INVERT transform of A034943 and the INVERTi transform of A094790. - Gary W. Adamson,  Apr 01 2011 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298. Guo-Niu Han, Enumeration of Standard Puzzles Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy] M. M. Mogbonju, I. A. Ogunleke, O. A. Ojo, Graphical Representation Of Conjugacy Classes In The Order-Preserving Full Transformation Semigroup, International Journal of Scientific Research and Engineering Studies (IJSRES), Volume 1 Issue 5, November 2014; ISSN: 2349-8862. László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (1st line of Table 1 is 3*a(n-2)). László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (1st line of Table 1 is a(n-2)). Yan X Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015. Index entries for linear recurrences with constant coefficients, signature (4,-4,1). FORMULA a(n) = ((((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5)) + 1. a(n) = Sum_{m=0..n} A055587(n, m) = 1 + A001906(n). G.f.: (1-2*x)/((1 - 3*x + x^2)*(1-x)). From Paul Barry, Oct 07 2004: (Start) a(n) = 4a(n-1) - 4a(n-2) + a(n-3); a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2k)2^(n-3k). (End) From Paul Barry, Oct 26 2004: (Start) a(n) = Fibonacci(2n) + 1; a(n) = Sum_{k=0..n} Fibonacci(2k+2)(2*0^(n-k)-1); a(n) = A008346(2n). (End) a(n) = Sum_{k=0..2n+1} ((-1)^(k+1))*fibonacci(k). - Michel Lagneau, Feb 03 2014 MAPLE g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)+1), n=0..27); # Zerinvary Lajos, Mar 22 2009 MATHEMATICA Join[{a=1, b=2}, Table[c=3*b-1*a-1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2011 *) Table[Fibonacci[2 n] + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -4, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Sep 30 2017 *) PROG (Sage) [lucas_number1(n, 3, 1)+1 for n in range(29)] # Zerinvary Lajos, Jul 06 2008 (MAGMA) [Fibonacci(2*n)+1: n in [0..30]]; // Vincenzo Librandi, Sep 30 2017 CROSSREFS Cf. A055587, A001906. Partial sums of A001519. Apart from first term, same as A052925. Cf. A144955. Cf. A034943, A094790. Sequence in context: A115324 A196307 A107092 * A088456 A091561 A025265 Adjacent sequences:  A055585 A055586 A055587 * A055589 A055590 A055591 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, May 30 2000; Barry E. Williams, Jun 04 2000 STATUS approved

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