A181214 Number of n X 3 binary matrices with no three 1's adjacent in a line diagonally or antidiagonally.

8, 64, 400, 2500, 16100, 103684, 665252, 4268356, 27399292, 175880644, 1128941012, 7246435876, 46513697660, 298563888100, 1916431442740, 12301251494596, 78959676072668, 506828955431044, 3253250254953428, 20882069005614436

Offset

1,1

Comments

Column 3 of A181217.

Links

R. H. Hardin, Table of n, a(n) for n = 1..300

Robert Israel, Maple-assisted proof of formula

Index entries for linear recurrences with constant coefficients, signature (6,0,16, 21,-78,-32,0,-12,8).

Formula

Empirical: a(n) = 6*a(n-1) + 16*a(n-3) + 21*a(n-4) - 78*a(n-5) - 32*a(n-6) - 12*a(n-8) + 8*a(n-9).

Empirical g.f.: 4*x*(2 + 4*x + 4*x^2 - 7*x^3 - 23*x^4 - 9*x^5 - x^6 - 2*x^7 + 2*x^8) / ((1 - 6*x - 3*x^2 + 2*x^3)*(1 + 3*x^2 - 12*x^4 - 4*x^6)). - Colin Barker, Feb 22 2018

Empirical formula confirmed by Robert Israel, Apr 30 2018: see link.

Example

Some avoided solutions for 4 X 3:
  0 0 1   0 0 0   0 0 0   1 0 0   0 0 0   0 0 0   0 0 0
  0 0 1   0 0 1   1 0 1   0 0 1   1 0 1   1 0 0   0 0 1
  0 1 0   1 1 0   0 1 0   0 1 0   0 1 0   0 1 1   0 1 0
  1 0 0   1 0 0   0 0 1   1 0 0   1 0 0   0 0 1   1 0 1

Maple

f:= gfun:-rectoproc({a(n) = 6*a(n-1) + 16*a(n-3) + 21*a(n-4) - 78*a(n-5) - 32*a(n-6) - 12*a(n-8) + 8*a(n-9), seq(a(i)=[8, 64, 400, 2500, 16100, 103684, 665252, 4268356, 27399292][i], i=1..9)}, a(n), remember):
map(f, [$1..20]); # Robert Israel, Apr 30 2018

Mathematica

LinearRecurrence[{6, 0, 16, 21, -78, -32, 0, -12, 8}, {8, 64, 400, 2500, 16100, 103684, 665252, 4268356, 27399292}, 20] (* Vincenzo Librandi, May 01 2018 *)

Prog

(Magma) I:=[8, 64, 400, 2500, 16100, 103684, 665252, 4268356, 27399292]; [n le 9 select I[n] else 6*Self(n-1)+16*Self(n-3)+21*Self(n-4)-78*Self(n-5) -32*Self(n-6)-12*Self(n-8)+8*Self(n-9): n in [1..25]]; // Vincenzo Librandi, May 01 2018

Crossrefs

Cf. A181217.

Sequence in context: A293888 A218267 A189105 * A283859 A222446 A239034

Adjacent sequences: A181211 A181212 A181213 * A181215 A181216 A181217

Keyword

nonn

Author

R. H. Hardin, Oct 10 2010

Status

approved