A100313 Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).

1, 16, 96, 400, 1408, 4480, 13312, 37632, 102400, 270336, 696320, 1757184, 4358144, 10649600, 25690112, 61276160, 144703488, 338690048, 786432000, 1812987904, 4152360960, 9453961216, 21407727616, 48234496000, 108179488768, 241591910400, 537407782912

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 (8,-24,32,-16).

Crossrefs

Cf. A055585, A100312 (m=3), this sequence (m=4).

Programs

Magma

[2^n*n*(n^2+9*n+14)/3 +0^n: n in [0..40]]; // G. C. Greubel, Feb 01 2023

Mathematica

Table[If[n==0, 1, 2^n*n*(n^2+9*n+14)/3], {n,0,40}] (* G. C. Greubel, Feb 01 2023 *)

PARI

vector(50, n, n*(n^2+9*n+14) * 2^n / 3) \\ Michel Marcus, Dec 01 2014

SageMath

[2^n*n*(n^2+9*n+14)/3 +0^n for n in range(41)] # G. C. Greubel, Feb 01 2023

Formula

G.f.: 1 + 16*x*(1-x)^2/(1-2*x)^4.

a(n) = (1/3) n*(n^2 + 9*n + 14) * 2^n for n>0, with a(0) = 1.

a(n) = 16 * A055585(n-1) for n>0.

E.g.f.: (1/3)*(3 + 8*x*(6 + 6*x + x^2)*exp(2*x)). - G. C. Greubel, Feb 01 2023

Extensions

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