A295035 - OEIS

%I #11 Nov 13 2017 06:28:28

%S 4,29,140,741,3853,19956,104096,541775,2819027,14671776,76356631,

%T 397392690,2068192977,10763659993,56018290276,291541126302,

%U 1517293977155,7896590852333,41096944195828,213884557644749,1113138825391146

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

%C Column 3 of A295040.

%H R. H. Hardin, <a href="/A295035/b295035.txt">Table of n, a(n) for n = 1..210</a>

%H Robert Israel, <a href="/A295035/a295035_1.pdf">Maple-assisted proof of formula</a>

%F 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)

%F Empirical formula confirmed by _Robert Israel_, Nov 12 2017 (see link).

%e Some solutions for n=7

%e ..1..1..0. .1..0..0. .0..0..0. .0..0..1. .0..1..0. .1..1..0. .0..0..0

%e ..0..0..0. .1..0..0. .1..1..1. .1..0..1. .0..1..0. .1..0..1. .0..0..1

%e ..1..1..0. .1..1..0. .0..0..1. .1..1..0. .0..1..0. .0..1..1. .0..1..1

%e ..0..0..0. .0..0..1. .0..0..0. .0..0..0. .0..1..0. .0..1..0. .1..1..0

%e ..0..0..0. .0..0..1. .1..1..0. .1..1..1. .1..0..1. .0..1..1. .1..0..1

%e ..1..1..0. .1..1..0. .0..1..0. .0..0..0. .1..0..1. .1..0..1. .0..0..1

%e ..1..0..0. .0..0..0. .0..0..0. .0..0..0. .1..1..1. .1..0..1. .0..0..1

%p q:= proc(a,b) local r,s,t,M,i;

%p    s:= floor((a-1)/8);

%p    if s <> (b-1) mod 8 then return 0 fi;

%p    s:= convert(s+8,base,2);

%p    r:= convert(8+floor((b-1)/8),base,2);

%p    t:= convert(8+ ((a-1) mod 8),base,2);

%p    M:= Vector(3);

%p    if s[1] = 1 and s[2] = 1 then M[1]:= 1; M[2]:= 1 fi;

%p    if s[2]=1 and s[3]=1 then M[2]:= M[2]+1; M[3]:= 1 fi;

%p    for i from 1 to 3 do if s[i]=1 then

%p      M[i]:= M[i]+r[i]+t[i];

%p      if M[i] = 0 or M[i]>2 then return 0 fi;

%p    fi od;

%p    1

%p end proc:

%p T:= Matrix(64,64, q);

%p u:= Vector[row](64):

%p v:= Vector(64):

%p for i from 0 to 7 do u[8*i+1]:= 1; v[i+1]:= 1;

%p od:

%p seq(u . T^n . v, n = 1 .. 100); # _Robert Israel_, Nov 12 2017

%Y Cf. A295040.

%K nonn

%O 1,1

%A _R. H. Hardin_, Nov 12 2017