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A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

%I A100314 #46 Feb 01 2023 16:42:49
%S A100314 1,4,8,14,24,42,76,142,272,530,1044,2070,4120,8218,16412,32798,65568,
%T A100314 131106,262180,524326,1048616,2097194,4194348,8388654,16777264,
%U A100314 33554482,67108916,134217782,268435512,536870970,1073741884,2147483710,4294967360,8589934658
%N A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
%C A100314 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).
%D A100314 Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.
%H A100314 Vincenzo Librandi, <a href="/A100314/b100314.txt">Table of n, a(n) for n = 0..2000</a>
%H A100314 Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/">Encyclopaedia of Cubature Formulas</a>
%H A100314 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 A100314 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A100314 a(n) = 2^n + 2*n.
%F A100314 From _Gary W. Adamson_, Jul 20 2007: (Start)
%F A100314 Binomial transform of (1, 3, 1, 1, 1, ...).
%F A100314 For n > 0, a(n) = 2*A005126(n-1). (End)
%F A100314 From _R. J. Mathar_, Jun 13 2008: (Start)
%F A100314 G.f.: 1 + 2*x*(2 -4*x +x^2)/((1-x)^2*(1-2*x)).
%F A100314 a(n+1)-a(n) = A052548(n). (End)
%F A100314 From _Colin Barker_, Oct 16 2013: (Start)
%F A100314 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
%F A100314 G.f.: (1 - 3*x^2)/((1-x)^2*(1-2*x)). (End)
%F A100314 E.g.f.: exp(2*x) + 2*x*exp(x). - _Franck Maminirina Ramaharo_, Dec 19 2018
%F A100314 a(n) = A000079(n) + A005843(n). - _Muniru A Asiru_, Dec 21 2018
%p A100314 a:= proc(n) 2^n + 2*n: end: seq(a(n),n=0..50); # _Gary W. Adamson_, Jul 20 2007
%t A100314 LinearRecurrence[{4,-5,2}, {1,4,8}, 34] (* _Jean-François Alcover_, Mar 19 2020 *)
%o A100314 (Magma) [2^n+2*n: n in [1..40]]; // _Vincenzo Librandi_, Oct 22 2011
%o A100314 (Maxima) makelist(2^n + 2*n, n, 0, 50); /* _Franck Maminirina Ramaharo_, Dec 19 2018 */
%o A100314 (GAP) List([0..40],n->2^n+2*n); # _Muniru A Asiru_, Dec 21 2018
%o A100314 (SageMath) [2^n +2*n for n in range(41)] # _G. C. Greubel_, Feb 01 2023
%Y A100314 Cf. this sequence (m=2), A100315 (m=3), A100316 (m=4).
%Y A100314 Cf. A000051, A000079, A001787, A002064, A005126, A005843.
%Y A100314 Cf. A052548, A058331, A099003, A132750, A176691.
%Y A100314 Row sums of A131830.
%K A100314 nonn,easy
%O A100314 0,2
%A A100314 _Sergey Kitaev_, Nov 13 2004
%E A100314 a(0)=1 prepended by _Alois P. Heinz_, Dec 21 2018