A350513 Number of 2 X n rectangular Celtic knots (up to rotations and symmetries).

3, 21, 648, 15552, 404595, 10812528, 290993472, 7849029276, 211855234563, 5719493462076, 154420985682648, 4169319089851512, 112571188876032435, 3039418274650387848, 82064259021919140432, 2215734684425617523796, 59824833697843168341123, 1615270484817096465311316

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Jessica Connor and Nick Ward, Celtic Knot Theory.

Index entries for linear recurrences with constant coefficients, signature (36,-216,-972,6561).

Formula

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

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

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.

G.f.: 3*x*(1 - 29*x + 180*x^2 - 108*x^3 - 4860*x^4 + 32805*x^5)/((1 - 9*x)*(1 - 27*x)*(1 - 27*x^2)). - Andrew Howroyd, Nov 15 2025

E.g.f.: (3*exp(9*x) + exp(27*x) + 12*cosh(3*sqrt(3)*x) + 6*sqrt(3)*sinh(3*sqrt(3)*x) - 16 - 270*x^2)/36. - Stefano Spezia, Nov 16 2025

Prog

(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

Crossrefs

Cf. A032120 (1 X n Celtic knots).

Sequence in context: A181003 A093549 A012044 * A098918 A001699 A291967

Adjacent sequences: A350510 A350511 A350512 * A350514 A350515 A350516

Keyword

nonn,easy

Author

Philippe Gibone, Jan 02 2022

Status

approved