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
Keywords
Examples
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
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).
Crossrefs
Cf. A181217.
Programs
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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
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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 *)
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.
Comments