login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

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 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 3 14:49 EST 2022. Contains 358534 sequences. (Running on oeis4.)