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Number of n X 3 0..1 arrays with every one equal to some NW, E or S neighbor.
1

%I #23 Aug 26 2022 13:19:37

%S 1,14,93,494,2801,16062,91161,517646,2942681,16724094,95039817,

%T 540117614,3069530009,17444277726,99136698537,563399082830,

%U 3201826086713,18196142535870,103409617841865,587682198231086,3339828270813785

%N Number of n X 3 0..1 arrays with every one equal to some NW, E or S neighbor.

%H R. H. Hardin, <a href="/A202901/b202901.txt">Table of n, a(n) for n = 1..210</a>

%H Robert Israel, <a href="/A202901/a202901_1.pdf">Maple-assisted proof of formula</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-17,32,-70,72,-24).

%F Empirical: a(n) = 8*a(n-1) -17*a(n-2) +32*a(n-3) -70*a(n-4) +72*a(n-5) -24*a(n-6).

%F Formula verified by _Robert Israel_, May 09 2018 (see link).

%F G.f.: x*(1 + 6*x - 2*x^2 - 44*x^3 + 52*x^4 - 16*x^5) / ((1 - x)*(1 - 7*x + 10*x^2 - 22*x^3 + 48*x^4 - 24*x^5)). - _Colin Barker_, Jun 02 2018

%e Some solutions for n=5:

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

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

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

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

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

%p f:= gfun:-rectoproc({a(n) = 8*a(n-1) -17*a(n-2) +32*a(n-3) -70*a(n-4) +72*a(n-5) -24*a(n-6), seq(a(i)=[1,14,93,494,2801,16062][i],i=1..6)},

%p a(n), remember):

%p map(f, [$1..25]); # _Robert Israel_, May 09 2018

%t LinearRecurrence[{8, -17, 32, -70, 72, -24}, {1, 14, 93, 494, 2801, 16062}, 21] (* _Jean-François Alcover_, Aug 26 2022 *)

%Y Column 3 of A202906.

%K nonn,easy

%O 1,2

%A _R. H. Hardin_, Dec 25 2011