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A185830 Half the number of n X 4 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors 1
2, 23, 118, 514, 2398, 11789, 54223, 257050, 1213538, 5716561, 26960702, 127201987, 599792318, 2828918061, 13342117403, 62924057051, 296766436047, 1399631468891, 6601012746804, 31132105093032, 146827124366034, 692474808791206 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Column 4 of A185835.

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..200

Robert Israel, Maple-assisted proof of formula

FORMULA

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).

Empirical formula verified (see link). - Robert Israel, Aug 15 2018

EXAMPLE

Some solutions for 6 X 4 with a(1,1)=0:

  0 0 0 0    0 0 1 1    0 0 1 0    0 1 1 1    0 1 1 1

  0 1 1 1    1 0 1 0    1 1 1 0    0 1 0 0    0 1 0 0

  0 1 0 1    1 1 0 0    1 0 0 1    0 1 1 0    1 1 0 1

  0 1 0 1    0 0 1 1    0 1 1 1    1 0 1 0    1 0 0 1

  0 1 0 1    0 1 0 1    0 0 0 0    1 0 1 0    0 1 1 0

  0 0 1 1    0 1 0 1    1 1 1 0    1 1 1 0    0 1 1 0

MAPLE

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,

[seq(convert(2^8+i, base, 2)[1..8], i=0..2^8-1)]):

Compatible:= proc(i, j) local k;

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})

   or not member(numboccur(Configs[i][6], [Configs[i][2], Configs[i][5], Configs[i][7], Configs[j][6]]), {1, 2})

   or not member(numboccur(Configs[i][7], [Configs[i][3], Configs[i][6], Configs[i][8], Configs[j][7]]), {1, 2})

   or not member(numboccur(Configs[i][8], [Configs[i][4], Configs[i][7], Configs[j][8]]), {1, 2})

  then 0 else 1 fi;

end proc:

T:= Matrix(162, 162, Compatible):

u:= Vector(162, proc(i) if member(numboccur(Configs[i][1], [Configs[i][2], Configs[i][5]]), {1, 2})

    and member(numboccur(Configs[i][2], [Configs[i][1], Configs[i][3], Configs[i][6]]), {1, 2})

    and member(numboccur(Configs[i][3], [Configs[i][2], Configs[i][4], Configs[i][7]]), {1, 2})

    and member(numboccur(Configs[i][4], [Configs[i][3], Configs[i][8]]), {1, 2}) then 1 else 0 fi end proc) :

v:= Vector(162, proc(i) if member(numboccur(Configs[i][5], [Configs[i][1], Configs[i][6]]), {1, 2})

    and member(numboccur(Configs[i][6], [Configs[i][2], Configs[i][5], Configs[i][7]]), {1, 2})

    and member(numboccur(Configs[i][7], [Configs[i][3], Configs[i][6], Configs[i][8]]), {1, 2})

    and member(numboccur(Configs[i][8], [Configs[i][4], Configs[i][7]]), {1, 2}) then 1 else 0 fi end proc) :

Tv[0]:= v:

for n from 1 to 50 do Tv[n]:= T . Tv[n-1] od:

[2, seq(u^%T . Tv[n]/2, n=0..50)]; # Robert Israel, Aug 15 2018

CROSSREFS

Cf. A185835.

Sequence in context: A229222 A143912 A041579 * A301665 A193981 A235594

Adjacent sequences:  A185827 A185828 A185829 * A185831 A185832 A185833

KEYWORD

nonn

AUTHOR

R. H. Hardin, Feb 05 2011

STATUS

approved

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Last modified December 11 18:34 EST 2019. Contains 329925 sequences. (Running on oeis4.)