A193168 - OEIS

%I #20 Nov 06 2019 12:39:51

%S 1,24,235,2922,38169,494596,6375379,82191766,1059980385,13670322763,

%T 176299392634,2273637717194,29321902354841,378149186319554,

%U 4876791443282017,62893416210079645,811103334429880838,10460373436120693109,134901938876902858230,1739759409373842739031

%N Number of simple paths from (1, 1) to (n, 3) on an n X 3 grid with king moves allowed.

%C Simple paths are self-avoiding walks.

%C Note: Number of paths from (1, 1) to (n, 2) on a n X 2 grid matches the sequence A086347.

%C Example: a(2) = 24 paths from (1, 1) to (2, 3)

%C 1 [(1, 1), (2, 1), (1, 2), (1, 3), (2, 3)]

%C 2 [(1, 1), (2, 2), (1, 3), (2, 3)]

%C 3 [(1, 1), (2, 2), (2, 1), (1, 2), (2, 3)]

%C 4 [(1, 1), (1, 2), (2, 1), (2, 2), (1, 3), (2, 3)]

%C 5 [(1, 1), (2, 2), (2, 1), (1, 2), (1, 3), (2, 3)]

%C 6 [(1, 1), (1, 2), (2, 2), (1, 3), (2, 3)]

%C 7 [(1, 1), (1, 2), (1, 3), (2, 3)]

%C 8 [(1, 1), (2, 2), (1, 3), (1, 2), (2, 3)]

%C 9 [(1, 1), (2, 2), (2, 3)]

%C 10 [(1, 1), (1, 2), (2, 3)]

%C 11 [(1, 1), (2, 2), (1, 2), (1, 3), (2, 3)]

%C 12 [(1, 1), (2, 1), (2, 2), (1, 3), (2, 3)]

%C 13 [(1, 1), (2, 1), (2, 2), (1, 2), (2, 3)]

%C 14 [(1, 1), (2, 1), (1, 2), (1, 3), (2, 2), (2, 3)]

%C 15 [(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)]

%C 16 [(1, 1), (2, 1), (2, 2), (1, 3), (1, 2), (2, 3)]

%C 17 [(1, 1), (2, 1), (1, 2), (2, 3)]

%C 18 [(1, 1), (1, 2), (2, 2), (2, 3)]

%C 19 [(1, 1), (1, 2), (1, 3), (2, 2), (2, 3)]

%C 20 [(1, 1), (2, 1), (2, 2), (1, 2), (1, 3), (2, 3)]

%C 21 [(1, 1), (2, 2), (1, 2), (2, 3)]

%C 22 [(1, 1), (2, 1), (2, 2), (2, 3)]

%C 23 [(1, 1), (2, 1), (1, 2), (2, 2), (2, 3)]

%C 24 [(1, 1), (2, 1), (1, 2), (2, 2), (1, 3), (2, 3)]

%H Andrew Howroyd, <a href="/A193168/b193168.txt">Table of n, a(n) for n = 1..200</a>

%F Conjectures from _Andrew Howroyd_, Nov 05 2019: (Start)

%F a(n) = 12*a(n-1) + 6*a(n-2) + 50*a(n-3) + 300*a(n-4) - 194*a(n-5) - 833*a(n-6) + 352*a(n-7) + 661*a(n-8) - 84*a(n-9) - 219*a(n-10) + 72*a(n-11) + 54*a(n-12) for n > 12.

%F G.f.: x*(1 + 12*x - 59*x^2 - 92*x^3 + 195*x^4 + 280*x^5 + 102*x^6 - 178*x^7 - 67*x^8 + 109*x^9 - 75*x^10 - 54*x^11)/((1 + x + 9*x^2 + 24*x^3 + 9*x^4)*(1 - 13*x - 2*x^2 + 45*x^3 - 24*x^4 - 22*x^5 + 9*x^6 + 8*x^7 - 6*x^8)).

%F (End)

%Y Row 3 of A329118.

%Y Cf. A086347, A140518.

%K nonn,walk

%O 1,2

%A _Matías Benzo_, Jul 17 2011

%E Offset changed and more terms from _Andrew Howroyd_, Nov 05 2019