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%I #24 Feb 01 2023 16:48:23
%S 1,8,14,22,34,54,90,158,290,550,1066,2094,4146,8246,16442,32830,65602,
%T 131142,262218,524366,1048658,2097238,4194394,8388702,16777314,
%U 33554534,67108970,134217838,268435570,536871030,1073741946,2147483774,4294967426,8589934726
%N Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
%C An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).
%H G. C. Greubel, <a href="/A100315/b100315.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F a(n) = 2^n + 4*n + 2 for n>0, a(0)=1.
%F From _Chai Wah Wu_, Aug 26 2016: (Start)
%F a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3.
%F G.f.: 1 + 2*x*(4 - 9*x + 3*x^2)/((1-x)^2*(1-2*x)). (End)
%F E.g.f.: exp(2*x) + 2*(1+2*x)*exp(x) - 2. - _G. C. Greubel_, Feb 01 2023
%t Table[If[n==0, 1, 2^n+4*n+2], {n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011 *)
%o (Magma) [2^n+4*n+2*(1-0^n): n in [0..40]]; // _G. C. Greubel_, Feb 01 2023
%o (SageMath) [2^n+4*n+2*(1-0^n) for n in range(41)] # _G. C. Greubel_, Feb 01 2023
%Y Cf. A100314 (m=2), this sequence (m=3), A100316 (m=4).
%K nonn,easy
%O 0,2
%A _Sergey Kitaev_, Nov 13 2004
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 21 2018
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