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

%I #26 Sep 08 2022 08:46:04

%S 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,

%T 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,

%U 3,2,3,1,4,1,3,2,3,1,4,1,3,2,3,1,4,1,3

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

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

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

%H Alois P. Heinz, <a href="/A220128/b220128.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1,1).

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

%F From _Wesley Ivan Hurt_, Jun 20 2016: (Start)

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

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

%F 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

%e a(6) = 4, because there are 4 tilings of a 6 X 3 rectangle using integer-sided rectangular tiles of area 6:

%e ._._._. .___._. ._.___. ._____.

%e | | | | | | | | | | | |

%e | | | | | | | | | | |_____|

%e | | | | |___| | | |___| | |

%e | | | | | | | | | | |_____|

%e | | | | | | | | | | | |

%e |_|_|_| |___|_| |_|___| |_____|

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

%t PadRight[{1}, 120, {4,1,3,2,3,1}] (* _Harvey P. Dale_, Jan 06 2016 *)

%o (Magma) [1] cat &cat [[1, 3, 2, 3, 1, 4]^^20]; // _Wesley Ivan Hurt_, Jun 20 2016

%Y Row n=3 of A220122.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Dec 06 2012

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)