A295035 Number of nX3 0..1 arrays with each 1 horizontally or vertically adjacent to 1 or 2 1s.

4, 29, 140, 741, 3853, 19956, 104096, 541775, 2819027, 14671776, 76356631, 397392690, 2068192977, 10763659993, 56018290276, 291541126302, 1517293977155, 7896590852333, 41096944195828, 213884557644749, 1113138825391146

Offset: 1

Author

R. H. Hardin, Nov 12 2017

Keywords

• nonn

Comments

Column 3 of A295040.

Examples

			Some solutions for n=7
..1..1..0. .1..0..0. .0..0..0. .0..0..1. .0..1..0. .1..1..0. .0..0..0
..0..0..0. .1..0..0. .1..1..1. .1..0..1. .0..1..0. .1..0..1. .0..0..1
..1..1..0. .1..1..0. .0..0..1. .1..1..0. .0..1..0. .0..1..1. .0..1..1
..0..0..0. .0..0..1. .0..0..0. .0..0..0. .0..1..0. .0..1..0. .1..1..0
..0..0..0. .0..0..1. .1..1..0. .1..1..1. .1..0..1. .0..1..1. .1..0..1
..1..1..0. .1..1..0. .0..1..0. .0..0..0. .1..0..1. .1..0..1. .0..0..1
..1..0..0. .0..0..0. .0..0..0. .0..0..0. .1..1..1. .1..0..1. .0..0..1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Robert Israel, Maple-assisted proof of formula

Crossrefs

Cf. A295040.

Programs

Maple

q:= proc(a,b) local r,s,t,M,i;
   s:= floor((a-1)/8);
   if s <> (b-1) mod 8 then return 0 fi;
   s:= convert(s+8,base,2);
   r:= convert(8+floor((b-1)/8),base,2);
   t:= convert(8+ ((a-1) mod 8),base,2);
   M:= Vector(3);
   if s[1] = 1 and s[2] = 1 then M[1]:= 1; M[2]:= 1 fi;
   if s[2]=1 and s[3]=1 then M[2]:= M[2]+1; M[3]:= 1 fi;
   for i from 1 to 3 do if s[i]=1 then
     M[i]:= M[i]+r[i]+t[i];
     if M[i] = 0 or M[i]>2 then return 0 fi;
   fi od;
   1
end proc:
T:= Matrix(64,64, q);
u:= Vector[row](64):
v:= Vector(64):
for i from 0 to 7 do u[8*i+1]:= 1; v[i+1]:= 1;
od:
seq(u . T^n . v, n = 1 .. 100); # Robert Israel, Nov 12 2017

Formula

Empirical: a(n) = 4*a(n-1) +2*a(n-2) +17*a(n-3) +18*a(n-4) +53*a(n-5) -23*a(n-6) -38*a(n-7) +52*a(n-8) -19*a(n-9) -14*a(n-10) +16*a(n-11) -11*a(n-12) +a(n-14) -a(n-15)

Empirical formula confirmed by Robert Israel, Nov 12 2017 (see link).