|
%I #24 Feb 09 2023 21:58:08
%S 1,16,24,34,48,70,108,178,312,574,1092,2122,4176,8278,16476,32866,
%T 65640,131182,262260,524410,1048704,2097286,4194444,8388754,16777368,
%U 33554590,67109028,134217898,268435632,536871094,1073742012,2147483842,4294967496,8589934798
%N Number of 4 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="/A100316/b100316.txt">Table of n, a(n) for n = 0..1000</a>
%H Sergey 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 + 6*n + 8 for n>0, a(0) = 1.
%F G.f.: (1+12*x-35*x^2+16*x^3)/((1-2*x)*(1-x)^2). - _Alois P. Heinz_, Dec 21 2018
%F E.g.f.: exp(2*x) + 2*(4+3*x)*exp(x) - 8. - _G. C. Greubel_, Feb 01 2023
%t Table[If[n==0, 1, 2^n+6*n+8], {n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011 *)
%o (Magma) [2^n+6*n+8*(1-0^n): n in [0..40]]; // _G. C. Greubel_, Feb 01 2023
%o (SageMath) [2^n+6*n+8*(1-0^n) for n in range(41)] # _G. C. Greubel_, Feb 01 2023
%Y Cf. A100314 (m=2), A100315 (m=3), this sequence (m=4).
%K nonn
%O 0,2
%A _Sergey Kitaev_, Nov 13 2004
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 21 2018
|
