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A295035 Number of nX3 0..1 arrays with each 1 horizontally or vertically adjacent to 1 or 2 1s. 1
4, 29, 140, 741, 3853, 19956, 104096, 541775, 2819027, 14671776, 76356631, 397392690, 2068192977, 10763659993, 56018290276, 291541126302, 1517293977155, 7896590852333, 41096944195828, 213884557644749, 1113138825391146 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Column 3 of A295040.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

Robert Israel, Maple-assisted proof of formula

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).

EXAMPLE

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

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

CROSSREFS

Cf. A295040.

Sequence in context: A095670 A273074 A247748 * A295526 A272373 A273247

Adjacent sequences:  A295032 A295033 A295034 * A295036 A295037 A295038

KEYWORD

nonn

AUTHOR

R. H. Hardin, Nov 12 2017

STATUS

approved

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Last modified October 25 09:41 EDT 2021. Contains 348239 sequences. (Running on oeis4.)