%I A030186
%S A030186 1,2,7,22,71,228,733,2356,7573,24342,78243,251498,808395,2598440,
%T A030186 8352217,26846696,86293865,277376074,891575391,2865808382,9211624463,
%U A030186 29609106380,95173135221,305916887580,983314691581,3160687827102
%N A030186 a(n) = 3a(n-1) + a(n-2) - a(n-3), n >= 3; a(0) = 1, a(1) = 2, a(2) = 
               7.
%C A030186 Number of matchings in ladder graph L_n = P_2 X P_n
%C A030186 Cycle-path coverings of a family of digraphs.
%C A030186 a(n+1) = Fibonacci(n+1)^2 + Sum_{k=0..n} Fibonacci(k)^2*a(n-k) (with 
               the offset convention Fibonacci(2)=2). - Barry Cipra (bcipra(AT)rconnect.com), 
               Jun 11 2003
%C A030186 Equivalently, ways of paving a 2xn grid cells using only singletons and 
               dominoes. - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 25 2005
%C A030186 It is easy to see that the g.f. for indecomposable tilings (pavings) 
               i.e. those that cannot be split vertically into smaller tilings, 
               is g=2x+3x^2+2x^3+2x^4+2x^5+...=x(2+x-x^2)/(1-x); then G.f.=1/(1-g)=(1-x)/
               (1-3x-x^2+x^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 
               16 2006
%C A030186 Row sums of A046741. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 
               16 2006
%C A030186 Equals binomial transform of A156096 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Feb 03 2009]
%D A030186 O. M. D'Antona and E. Munarini, The Cycle-path Indicator Polynomial of 
               a Digraph, Advances in Applied Mathematics 25 (2000), 41-56.
%D A030186 R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. 
               Phys., 15 (1974), 214-216.
%D A030186 J. D. E. Konhauser et al., Which Way Did The Bicycle Go? Problem 140 
               "Count The Tilings" pp. 42; 180-1 Dolciani Math. Exp. No. 18 MAA 
               Washington DC 1996.
%D A030186 Per Hakan Lundow, "Computation of matching polynomials and the number 
               of 1-factors in polygraphs", Research reports, No 12, 1996, Department 
               of Mathematics, Umea University.
%D A030186 R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x 
               N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
%H A030186 T. D. Noe, <a href="b030186.txt">Table of n, a(n) for n=0..200</a>
%H A030186 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=473">
               Encyclopedia of Combinatorial Structures 473</a>
%H A030186 Per Hakan Lundow, <a href="http://www.theophys.kth.se/~phl/Text/1factors2.ps.gz">
               Enumeration of matchings in polygraphs</a>, 1998.
%H A030186 N. J. A. Sloane <a href="a030186.txt">Notes on A030186 and A033505</a>
%F A030186 G.f.: (1-t)/(1-3t-t^2+t^3).
%p A030186 a[0]:=1: a[1]:=2: a[2]:=7: for n from 3 to 25 do a[n]:=3*a[n-1]+a[n-2]-a[n-3] 
               od: seq(a[n],n=0..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 16 2006
%Y A030186 Partial sums give A033505.
%Y A030186 Cf. A054894, A055518, A055519.
%Y A030186 A156096 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 03 2009]
%Y A030186 Sequence in context: A106438 A109999 A092690 this_sequence A162770 A116387 
               A114495
%Y A030186 Adjacent sequences: A030183 A030184 A030185 this_sequence A030187 A030188 
               A030189
%K A030186 nonn,easy,nice
%O A030186 0,2
%A A030186 Ottavio D'Antona (dantona(AT)dsi.unimi.it)
%E A030186 More terms from James A. Sellers (sellersj(AT)math.psu.edu)
%E A030186 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Feb 13, 
               2002.