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

0,3

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... .

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).

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

Example

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

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 *)

Prog

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

Crossrefs

Row n=3 of A220122.

Sequence in context: A106584 A213940 A236027 * A324468 A131498 A236455

Adjacent sequences: A220125 A220126 A220127 * A220129 A220130 A220131

Keyword

nonn,easy

Author

Alois P. Heinz, Dec 06 2012

Status

approved