A100312 Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).

1, 8, 32, 104, 304, 832, 2176, 5504, 13568, 32768, 77824, 182272, 421888, 966656, 2195456, 4947968, 11075584, 24641536, 54525952, 120061952, 263192576, 574619648, 1249902592, 2709520384, 5855248384, 12616466432, 27111981056, 58116276224, 124285616128

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 the g.f. 2*x*y/(1-2*(x+y-x*y)).

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 (6,-12,8).

Crossrefs

Cf. A049611, this sequence (m=3), A100313 (m=4).

Programs

Magma

[2^(n-1)*(n^2+5*n+2): n in [0..50]]; // G. C. Greubel, Feb 01 2023

Mathematica

Table[2^(n-1)*(n^2+5*n+2), {n,0,50}] (* G. C. Greubel, Feb 01 2023 *)

PARI

vector(50, n, (n^2 + 5*n + 2) * 2^(n-1)) \\ Michel Marcus, Dec 01 2014

SageMath

[2^(n-1)*(n^2+5*n+2) for n in range(51)] # G. C. Greubel, Feb 01 2023

Formula

G.f.: 1 + 8*x*(1-x)^2/(1-2*x)^3.

a(n) = 2^(n-1) * (n^2 + 5*n + 2).

a(n) = 8 * A049611(n) for n>0.

E.g.f.: (1 + 6*x + 2*x^2)*exp(2*x). - G. C. Greubel, Feb 01 2023

Extensions

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