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%I A022026 #42 Aug 06 2022 07:20:34
%S A022026 2,15,112,836,6240,46576,347648,2594880,19368448,144568064,1079070720,
%T A022026 8054293504,60118065152,448727347200,3349346516992,24999862747136,
%U A022026 186601515909120,1392812676284416,10396095346638848,77597512067973120,579195715157229568
%N A022026 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(2,15).
%C A022026 From _Alois P. Heinz_, Mar 18 2009: (Start)
%C A022026 a(n) is also the number of forests in the 2 X (n+1) grid.
%C A022026 a(0)=2, because there are 2 forests in the 2 X 1 grid: 1.2 and 1-2.
%C A022026 a(1)=15, because there are 15 forests in the 2 X 2 grid:
%C A022026   1.2 1-2 1.2 1.2 1.2 1-2 1-2 1-2 1.2 1.2 1.2 1.2 1-2 1-2 1-2
%C A022026   . . . . . | . . | . . | . . | . . | | | | . | | | . | | . |
%C A022026   4.3 4.3 4.3 4-3 4.3 4.3 4-3 4.3 4-3 4.3 4-3 4-3 4-3 4.3 4-3
%C A022026 a(n) = 8a(n-1) - 4a(n-2) for n>=2, because each of the a(n-1) forests can be extended by 8 patterns:
%C A022026   .o -o .o -o .o -o .o -o
%C A022026   .. .. .. .. .| .| .| .|
%C A022026   .o .o -o -o .o .o -o -o
%C A022026 where 4a(n-2) of these are not forests, namely the extensions of a(n-2) forests by 4 patterns:
%C A022026   .o-o -o-o .o-o -o-o
%C A022026   .| | .| | .| | .| |
%C A022026   .o-o .o-o -o-o -o-o (End)
%H A022026 Alois P. Heinz, <a href="/A022026/b022026.txt">Table of n, a(n) for n = 0..1000</a>
%H A022026 M. Desjarlais and R. Molina, <a href="https://web.archive.org/web/20060904233419/http://othello.alma.edu/~molina/papers/pdf%20papers/spantree.pdf">Counting Spanning Trees in Grid Graphs</a>
%H A022026 Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
%H A022026 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-4).
%F A022026 G.f.: (2-x)/(1-8x+4x^2). - David Boyd and _Ralf Stephan_, Apr 15 2004
%F A022026 From _Peter Bala_, May 03 2014: (Start)
%F A022026 a(n) = sum of the entries in the 2 X 2 matrix A^n where A is the 2 X 2 matrix [4, 4; 3, 4].
%F A022026 a(n) = (1 + 7*sqrt(3)/12)*(4 + 2*sqrt(3))^n + (1 - 7*sqrt(3)/12)*(4 - 2*sqrt(3))^n. See Desjarlais and Molina. (End)
%F A022026 a(n+1) = ceiling(a(n)^2/a(n-1))-1 for all n > 0. - _M. F. Hasler_, Feb 10 2016
%p A022026 a:= n-> (Matrix([[15,2]]). Matrix([[8, 1], [-4, 0]])^n)[1, 2]:
%p A022026 seq(a(n), n=0..35);  # _Alois P. Heinz_, Mar 18 2009
%t A022026 CoefficientList[Series[(2-x)/(1-8*x+4*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, May 03 2014 *)
%o A022026 (PARI) a(n)=([15,2]*[8,1;-4,0]^n)[2] \\ _M. F. Hasler_, Feb 10 2016
%Y A022026 Equals A028859(2n+2)/4.
%Y A022026 Cf. A022018 - A022025, A022027 - A022032.
%K A022026 nonn,easy
%O A022026 0,1
%A A022026 _R. K. Guy_

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