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%I #13 Aug 27 2022 08:14:35
%S 1,3,8,14,25,53,111,217,426,860,1733,3453,6885,13791,27616,55198,
%T 110341,220737,441563,883037,1765930,3532004,7064241,14128249,
%U 28256121,56512619,113025848,226051086,452101185,904203357,1808408311,3616815025
%N Number of nX3 0..1 arrays with each 1 horizontally or vertically adjacent to 2 or 4 1s.
%C Column 3 of A295205.
%H R. H. Hardin, <a href="/A295200/b295200.txt">Table of n, a(n) for n = 1..210</a>
%H Robert Israel, <a href="/A295200/a295200.pdf">Maple-assisted proof of formula</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature(2,-1,2,1,-2)
%F Empirical: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) +a(n-4) -2*a(n-5).
%F From _Robert Israel_, Nov 19 2017: (Start)
%F Empirical formula is true (see link).
%F G.f.: (x+x^2+3*x^3-x^4-2*x^5)/(1-2*x+x^2-2*x^3-x^4+2*x^5). (End)
%e Some solutions for n=7
%e ..1..1..0. .0..0..0. .0..0..0. .0..1..1. .0..0..0. .1..1..1. .0..0..0
%e ..1..1..0. .0..1..1. .1..1..0. .0..1..1. .1..1..0. .1..0..1. .0..0..0
%e ..0..0..0. .1..1..1. .1..1..0. .0..0..0. .1..1..0. .1..0..1. .0..0..0
%e ..0..1..1. .1..1..0. .0..0..0. .0..0..0. .0..0..0. .1..0..1. .0..0..0
%e ..0..1..1. .0..0..0. .1..1..0. .0..0..0. .1..1..0. .1..0..1. .1..1..0
%e ..0..0..0. .0..1..1. .1..1..0. .0..0..0. .1..1..1. .1..1..1. .1..1..1
%e ..0..0..0. .0..1..1. .0..0..0. .0..0..0. .0..1..1. .0..0..0. .0..1..1
%p f:= gfun:-rectoproc({a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) +a(n-4) -2*a(n-5),a(1)=1, a(2)=3, a(3)=8, a(4)=14,a(5)=25},a(n),remember):
%p map(f, [$1..100]); # _Robert Israel_, Nov 19 2017
%t LinearRecurrence[{2, -1, 2, 1, -2}, {1, 3, 8, 14, 25}, 32] (* _Jean-François Alcover_, Aug 27 2022 *)
%Y Cf. A295205.
%K nonn
%O 1,2
%A _R. H. Hardin_, Nov 16 2017
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