login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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
OFFSET
1,1
COMMENTS
Column 4 of A185835.
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
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 05 2011
STATUS
approved