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%I #14 Aug 15 2018 22:26:08
%S 2,23,118,514,2398,11789,54223,257050,1213538,5716561,26960702,
%T 127201987,599792318,2828918061,13342117403,62924057051,296766436047,
%U 1399631468891,6601012746804,31132105093032,146827124366034,692474808791206
%N Half the number of n X 4 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors
%C Column 4 of A185835.
%H R. H. Hardin, <a href="/A185830/b185830.txt">Table of n, a(n) for n = 1..200</a>
%H Robert Israel, <a href="/A185830/a185830.pdf">Maple-assisted proof of formula</a>
%F Empirical: a(n) = 4*a(n-1) + 9*a(n-2) - 15*a(n-3) - 67*a(n-4) + 2*a(n-5) + 321*a(n-6) - 70*a(n-7) - 672*a(n-8) - 351*a(n-9) + 920*a(n-10) - 5077*a(n-11) + 733*a(n-12) + 28694*a(n-13) + 15849*a(n-14) - 28046*a(n-15) - 56805*a(n-16) + 89957*a(n-17) - 87270*a(n-18) + 85004*a(n-19) - 164885*a(n-20) + 188247*a(n-21) - 111655*a(n-22) + 89028*a(n-23) - 117971*a(n-24) + 115994*a(n-25) - 64896*a(n-26) + 52709*a(n-27) - 50206*a(n-28) + 31595*a(n-29) - 11233*a(n-30) + 1156*a(n-31) + 2585*a(n-32) - 5924*a(n-33) + 2841*a(n-34) - 817*a(n-35) + 361*a(n-36) - 42*a(n-37) - 6*a(n-38).
%F Empirical formula verified (see link). - _Robert Israel_, Aug 15 2018
%e Some solutions for 6 X 4 with a(1,1)=0:
%e 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1
%e 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0
%e 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 1
%e 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1
%e 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0
%e 0 0 1 1 0 1 0 1 1 1 1 0 1 1 1 0 0 1 1 0
%p Configs:= remove(t -> min(nops({t[1],t[2],t[3],t[6]}), nops({t[2],t[3],t[4],t[7]}), nops({t[2],t[5],t[6],t[7]}), nops({t[3],t[6],t[7],t[8]}))=1,
%p [seq(convert(2^8+i,base,2)[1..8],i=0..2^8-1)]):
%p Compatible:= proc(i,j) local k;
%p if Configs[i][5..8] <> Configs[j][1..4] or not member(numboccur(Configs[i][5], [Configs[i][1],Configs[i][6],Configs[j][5]]),{1,2})
%p or not member(numboccur(Configs[i][6], [Configs[i][2],Configs[i][5],Configs[i][7],Configs[j][6]]),{1,2})
%p or not member(numboccur(Configs[i][7], [Configs[i][3],Configs[i][6],Configs[i][8],Configs[j][7]]),{1,2})
%p or not member(numboccur(Configs[i][8], [Configs[i][4],Configs[i][7],Configs[j][8]]),{1,2})
%p then 0 else 1 fi;
%p end proc:
%p T:= Matrix(162,162,Compatible):
%p u:= Vector(162,proc(i) if member(numboccur(Configs[i][1],[Configs[i][2],Configs[i][5]]),{1,2})
%p and member(numboccur(Configs[i][2],[Configs[i][1],Configs[i][3],Configs[i][6]]),{1,2})
%p and member(numboccur(Configs[i][3],[Configs[i][2],Configs[i][4],Configs[i][7]]),{1,2})
%p and member(numboccur(Configs[i][4],[Configs[i][3],Configs[i][8]]),{1,2}) then 1 else 0 fi end proc) :
%p v:= Vector(162,proc(i) if member(numboccur(Configs[i][5],[Configs[i][1],Configs[i][6]]),{1,2})
%p and member(numboccur(Configs[i][6],[Configs[i][2],Configs[i][5],Configs[i][7]]),{1,2})
%p and member(numboccur(Configs[i][7],[Configs[i][3],Configs[i][6],Configs[i][8]]),{1,2})
%p and member(numboccur(Configs[i][8],[Configs[i][4],Configs[i][7]]),{1,2}) then 1 else 0 fi end proc) :
%p Tv[0]:= v:
%p for n from 1 to 50 do Tv[n]:= T . Tv[n-1] od:
%p [2, seq(u^%T . Tv[n]/2,n=0..50)]; # _Robert Israel_, Aug 15 2018
%Y Cf. A185835.
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 05 2011