A100315 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).

1, 8, 14, 22, 34, 54, 90, 158, 290, 550, 1066, 2094, 4146, 8246, 16442, 32830, 65602, 131142, 262218, 524366, 1048658, 2097238, 4194394, 8388702, 16777314, 33554534, 67108970, 134217838, 268435570, 536871030, 1073741946, 2147483774, 4294967426, 8589934726

Offset: 0

Author

Sergey Kitaev, Nov 13 2004

Keywords

• nonn
• easy

Comments

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).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A100314 (m=2), this sequence (m=3), A100316 (m=4).

Programs

Magma

[2^n+4*n+2*(1-0^n): n in [0..40]]; // G. C. Greubel, Feb 01 2023

Mathematica

Table[If[n==0, 1, 2^n+4*n+2], {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)

SageMath

[2^n+4*n+2*(1-0^n) for n in range(41)] # G. C. Greubel, Feb 01 2023

Formula

a(n) = 2^n + 4*n + 2 for n>0, a(0)=1.

From Chai Wah Wu, Aug 26 2016: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3.

G.f.: 1 + 2*x*(4 - 9*x + 3*x^2)/((1-x)^2*(1-2*x)). (End)

E.g.f.: exp(2*x) + 2*(1+2*x)*exp(x) - 2. - G. C. Greubel, Feb 01 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018