A100316 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).

1, 16, 24, 34, 48, 70, 108, 178, 312, 574, 1092, 2122, 4176, 8278, 16476, 32866, 65640, 131182, 262260, 524410, 1048704, 2097286, 4194444, 8388754, 16777368, 33554590, 67109028, 134217898, 268435632, 536871094, 1073742012, 2147483842, 4294967496, 8589934798

Offset: 0

Author

Sergey Kitaev, Nov 13 2004

Keywords

• nonn

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
• Sergey 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), A100315 (m=3), this sequence (m=4).

Programs

Magma

[2^n+6*n+8*(1-0^n): n in [0..40]]; // G. C. Greubel, Feb 01 2023

Mathematica

Table[If[n==0, 1, 2^n+6*n+8], {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)

SageMath

[2^n+6*n+8*(1-0^n) for n in range(41)] # G. C. Greubel, Feb 01 2023

Formula

a(n) = 2^n + 6*n + 8 for n>0, a(0) = 1.

G.f.: (1+12*x-35*x^2+16*x^3)/((1-2*x)*(1-x)^2). - Alois P. Heinz, Dec 21 2018

E.g.f.: exp(2*x) + 2*(4+3*x)*exp(x) - 8. - G. C. Greubel, Feb 01 2023

Extensions

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