A220128 1 followed by period 6: (1, 3, 2, 3, 1, 4) repeated; offset 0.

1, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 3

Offset: 0

Author

Alois P. Heinz, Dec 06 2012

Keywords

• nonn
• easy

Comments

Also the number of tilings of an n X 3 rectangle using integer-sided rectangular tiles of area n.

Also decimal expansion of 12443/109890 = 0.1132314132314... .

Examples

			a(6) = 4, because there are 4 tilings of a 6 X 3 rectangle using integer-sided rectangular tiles of area 6:
._._._.  .___._.  ._.___.  ._____.
| | | |  |   | |  | |   |  |     |
| | | |  |   | |  | |   |  |_____|
| | | |  |___| |  | |___|  |     |
| | | |  |   | |  | |   |  |_____|
| | | |  |   | |  | |   |  |     |
|_|_|_|  |___|_|  |_|___|  |_____|
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Crossrefs

Row n=3 of A220122.

Programs

Magma

[1] cat &cat [[1, 3, 2, 3, 1, 4]^^20]; // Wesley Ivan Hurt, Jun 20 2016

Maple

a:=n-> `if`(n=0, 1, [4, 1, 3, 2, 3, 1][irem(n, 6)+1]): seq(a(n), n=0..100);

Mathematica

PadRight[{1}, 120, {4,1,3,2,3,1}] (* Harvey P. Dale, Jan 06 2016 *)

Formula

G.f.: (-3*x^4-4*x^3-4*x^2-2*x-1) / (x^4+x^3-x-1).

From Wesley Ivan Hurt, Jun 20 2016: (Start)

a(n) + a(n-1) = a(n-3) + a(n-4) for n>4.

a(0) = 1, a(n) = (7 + 3*cos(n*Pi) + 2*cos(2*n*Pi/3))/3 for n>0. (End)

E.g.f.: 2*(-9/2 + cos(sqrt(3)*x/2)*exp(-x/2) + 2*sinh(x) + 5*cosh(x))/3. - Ilya Gutkovskiy, Jun 21 2016