OFFSET
1,2
COMMENTS
Also number of permutations of length n which avoid the patterns 312, 2341, 3412; or avoid the patterns 132, 1324, 3214, etc.
Except for the offset, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^3; see A291000. - Clark Kimberling, Aug 24 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Andrew M. Baxter and Lara K. Pudwell, Ascent sequences avoiding pairs of patterns, Elect. J. Comb. 22(1) (2015), #P1.58.
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
David Callan, Toufik Mansour, Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns, Pure Math. Appl. 27(1) (2018), 62-97.
Qi Liu, Sergey Kitaev, and Philip B. Zhang, Simultaneous avoidance of length-4 patterns in ascent sequences, arXiv:2604.06735 [math.CO], 2026. See p. 4 (Table 1).
Shuzhen Lv and Sergey Kitaev, Stoimenow matchings avoiding multiple Catalan patterns simultaneously, arXiv:2509.12726 [math.CO], 2025. See pp. 3, 9.
Toufik Mansour and Mark Shattuck, Avoidance of type (1,2) patterns by Catalan words, Turkish Journal of Analysis and Number Theory, May 2017. See item 1-23 in Table 1, p. 3.
Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.
Lara Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015.
Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, Slides, Permutation Patterns 2014, East Tennessee State University Jul 07 2014.
Index entries for linear recurrences with constant coefficients, signature (4,-5,3).
FORMULA
G.f.: -((2x^2-2x+1)x)/(3x^3-5x^2+4x-1).
Binomial transform of A000930 starting with offset 1: [1, 1, 2, 3, 4, 6, 9, ...]. - Gary W. Adamson, Oct 23 2007
MATHEMATICA
CoefficientList[Series[x*(1-2*x+2*x^2)/(1-4*x+5*x^2-3*x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 29 2017 *)
PROG
(PARI) x='x+O('x^50); Vec(x*(1-2*x+2*x^2)/(1-4*x+5*x^2-3*x^3)) \\ G. C. Greubel, Apr 29 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lara Pudwell, Feb 26 2006
EXTENSIONS
Edited by N. J. A. Sloane, Mar 16 2008
STATUS
approved