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Number of 0..2 arrays x(0..n-1) of n elements with each no smaller than the sum of its four previous neighbors modulo 3.
1

%I #14 Dec 12 2023 10:31:50

%S 3,6,12,24,48,98,199,400,800,1597,3188,6360,12679,25273,50376,100400,

%T 200077,398698,794502,1583212,3154828,6286514,12526942,24961994,

%U 49740765,99116372,197505241,393560803,784232662,1562708632,3113946596,6205036280

%N Number of 0..2 arrays x(0..n-1) of n elements with each no smaller than the sum of its four previous neighbors modulo 3.

%C Column 2 of A200469

%H R. H. Hardin, <a href="/A200463/b200463.txt">Table of n, a(n) for n = 1..210</a>

%F a(n) - 3*a(n + 1) + 3*a(n + 2) - a(n + 3) + 2*a(n + 4) - 2*a(n + 5) - 4*a(n + 6) + 7*a(n + 7) - 2*a(n + 9) + 2*a(n + 10) - 10*a(n + 11) + 11*a(n + 12) + 3*a(n + 13) - 9*a(n + 14) + 7*a(n + 15) - 17*a(n + 16) + 22*a(n + 17) - 20*a(n + 18) + 24*a(n + 19) - 24*a(n + 20) + 13*a(n + 21) - 27*a(n + 22) + 36*a(n + 23) - 13*a(n + 24) - a(n + 25) - 11*a(n + 26) - 14*a(n + 27) + 42*a(n + 28) - 17*a(n + 29) + 9*a(n + 30) - 36*a(n + 31) + 17*a(n + 32) + 28*a(n + 33) - 19*a(n + 34) - 4*a(n + 35) + 22*a(n + 36) - 47*a(n + 37) + 82*a(n + 38) - 113*a(n + 39) + 135*a(n + 40) - 111*a(n + 41) + 45*a(n + 42) - a(n + 43) - 6*a(n + 44) + 43*a(n + 45) - 47*a(n + 46) + 17*a(n + 47) - 34*a(n + 48) + 42*a(n + 50) - 18*a(n + 51) + 10*a(n + 52) - 25*a(n + 53) + 45*a(n + 54) - 30*a(n + 55) - 18*a(n + 56) + 22*a(n + 57) - 20*a(n + 58) + 5*a(n + 59) + 27*a(n + 60) - 20*a(n + 61) + 5*a(n + 62) - 8*a(n + 63) + 23*a(n + 64) - 31*a(n + 65) + 14*a(n + 66) + 6*a(n + 67) - 24*a(n + 68) + 20*a(n + 69) + 5*a(n + 71) - 9*a(n + 72) + 3*a(n + 73) - 4*a(n + 74) - 6*a(n + 75) + 12*a(n + 76) - 4*a(n + 77) + 2*a(n + 79) - 3*a(n + 80) + a(n + 81) = 0. - _Robert Israel_, Dec 11 2023

%e Some solutions for n=6

%e ..1....0....1....1....0....1....0....0....0....0....2....1....0....1....0....0

%e ..2....0....2....2....0....2....0....0....1....2....2....2....0....2....2....0

%e ..0....0....0....2....0....1....1....1....2....2....2....1....0....1....2....0

%e ..0....0....0....2....2....2....1....2....1....2....0....1....0....2....2....1

%e ..0....0....2....2....2....1....2....0....2....0....2....2....1....1....1....2

%e ..2....0....2....2....1....0....2....2....1....2....0....2....2....2....1....2

%p Configs:= [seq(convert(81+i,base,3)[1..4],i=0..80)]:

%p Compatible:= proc(i,j)

%p if not Configs[i][2..4]=Configs[j][1..3] then return 0 fi;

%p if convert(Configs[i],`+`) mod 3 <= Configs[j][4]

%p then 1 else 0

%p fi

%p end proc:

%p T:= Matrix(81,81,Compatible):

%p initconfigs:= select(t -> t[1] <= t[2] and t[1]+t[2] mod 3 <= t[3] and t[1]+t[2]+t[3] mod 3 <= t[4], Configs):

%p u0:= Vector(81, i -> `if`(member(Configs[i],initconfigs),1,0)):

%p e:= Vector(81,1):

%p 3, 6, 12, seq(u0^%T . T^i . e, i=0..30); # _Robert Israel_, Dec 11 2023

%K nonn

%O 1,1

%A _R. H. Hardin_, Nov 18 2011