A005770 Number of convex polygons of length 2n on square lattice whose leftmost bottom vertex and rightmost top vertex have the same x-coordinate.

1, 9, 55, 286, 1362, 6143, 26729, 113471, 473471, 1951612, 7974660, 32384127, 130926391, 527657073, 2121795391, 8518575466, 34162154550, 136893468863, 548253828965, 2194897467395, 8784784672511, 35153438973304, 140653028240520, 562719731644671

Offset: 5

Author

Simon Plouffe, N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (12,-55,120,-125,54,-8).

Crossrefs

Cf. A005436, A005768, A005769.

Programs

Maple

A005770:=(1-3*z+2*z**2+z**3)/(4*z-1)/(2*z-1)/(z**2-3*z+1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[x^5*(1-3*x+2*x^2+x^3)/((1 - 3*x + x^2)^2*(1 - 6*x + 8*x^2)),{x,0,28}],x] (* Stefano Spezia, Jun 04 2024 *)

Formula

a(n) = A005436(n) - A005768(n) - A005769(n).

G.f.: x^5*(1-3*x+2*x^2+x^3)/((1 - 3*x + x^2)^2*(1 - 6*x + 8*x^2)). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

a(n) = 12*a(n-1) - 55*a(n-2) + 120*a(n-3) - 125*a(n-4) + 54*a(n-5) - 8*a(n-6) for n > 8. - Stefano Spezia, Jun 04 2024

Extensions

Better description from Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

More terms from Sean A. Irvine, Aug 26 2016