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Number of 2 X n rectangular Celtic knots (up to rotations and symmetries).
0

%I #22 Feb 14 2022 00:18:19

%S 3,21,648,15552,404595,10812528,290993472,7849029276,211855234563,

%T 5719493462076,154420985682648,4169319089851512,112571188876032435,

%U 3039418274650387848,82064259021919140432,2215734684425617523796,59824833697843168341123,1615270484817096465311316

%N Number of 2 X n rectangular Celtic knots (up to rotations and symmetries).

%H Jessica Connor and Nick Ward, <a href="https://www.maths.ed.ac.uk/~v1ranick/knots/celtic.pdf">Celtic Knot Theory</a>

%F a(n) = (3^(3*n-2) + 3^((3*n-2)/2) + 3^(2*n-1) + 3^(3*n/2))/4 for even n > 2;

%F a(n) = (3^(3*n-2) + 3^((3*n-1)/2) + 3^(2*n-1) + 3^((3*n-1)/2))/4 for odd n.

%F a(n) = (3^(3*n-2) + 3^floor((3*n-1)/2) + 3^(2*n-1) + 3^floor(3*n/2))/4 for n <> 2.

%o (PARI) a(n) = if(n==2, 21, (3^(3*n-2) + 3^((3*n-1)\2) + 3^(2*n-1) + 3^(3*n\2))/4) \\ _Andrew Howroyd_, Jan 03 2022

%Y Cf. A032120 (1 X n Celtic knots).

%K nonn

%O 1,1

%A _Philippe Gibone_, Jan 02 2022