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 A165407 Expansion of 1/(1-x-x^3*c(x^3)), c(x) the g.f. of A000108. 3
 1, 1, 1, 2, 3, 4, 7, 11, 16, 27, 43, 65, 108, 173, 267, 440, 707, 1105, 1812, 2917, 4597, 7514, 12111, 19196, 31307, 50503, 80380, 130883, 211263, 337284, 548547, 885831, 1417582, 2303413, 3720995, 5965622, 9686617, 15652239, 25130844, 40783083 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Hankel transform is A010892(n+1). Row sums of A165408. Number of UF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are UF-equivalent iff the positions of pattern UF are identical in these paths. This also works for the pattern FU. -  Sergey Kirgizov, Apr 08 2018 LINKS Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018. J.-L. Baril, A. Petrossian, Equivalence Classes of Motzkin Paths Modulo a Pattern of Length at Most Two, J. Int. Seq. 18 (2015) 15.7.1 FORMULA G.f.: 2/(1-2x+sqrt(1-4x^3))=1/(1-x-x^3/(1-x^3/(1-x^3/(1-x^3/(1-.... (continued fraction); a(n) = Sum_{k=0..n} if(n<=3k, C((n+k)/2,k)*((3k-n)/2+1)(1+(-1)^(n-k))/(2(k+1)); a(n) = Sum_{k=0..n+1} F(n-k+1)*(0^k-A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3). Conjecture: (n+1)*a(n) -(n+1)*a(n-1) -(n+1)*a(n-2) +2*(7-2*n)*a(n-3) +2*(2*n-7)*a(n-4) +2*(2*n-7)*a(n-5)=0. - R. J. Mathar, Nov 15 2011 a(3*n) = A026726(n); a(3*n+1) = A026671(n); a(3*n+2) = A026674(n+1). - Philippe Deléham, Feb 01 2014 CROSSREFS Sequence in context: A222023 A221997 A221998 * A039897 A297789 A222122 Adjacent sequences:  A165404 A165405 A165406 * A165408 A165409 A165410 KEYWORD easy,nonn AUTHOR Paul Barry, Sep 17 2009 STATUS approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)