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%I #22 May 29 2018 09:52:25
%S 1,2,6,20,62,194,612,1922,6038,18980,59646,187442,589076,1851266,
%T 5817894,18283700,57459518,180575906,567489348,1783428098,5604714422,
%U 17613731780,55354032894,173959101458,546694927604,1718078222594,5399341807686,16968314698580
%N Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shape P.
%H Alois P. Heinz, <a href="/A247076/b247076.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,5).
%F G.f.: (x-1)*(x^2+x+1)/(5*x^3+2*x^2+2*x-1).
%F a(n) = 2*a(n-1)+2*a(n-2)+5*a(n-3) for n>3, a(0)=1; a(1)=2, a(2)=6, a(3)=20.
%e a(2) = 6:
%e ._______. ._______. ._______. ._______. ._______. ._______.
%e | | | | | | | | | | | | | ._| | | |_. |
%e | ._| ._| |_. |_. | | ._|_. | |_. | ._| |___| | | |___|
%e |_| |_| | | |_| |_| |_| | |_| | |_|_| | | |___| |___| |
%e | | | | | | | | | | | | | ._| | | |_. |
%e |___|___| |___|___| |___|___| |___|___| |_|_____| |_____|_| .
%p a:= proc(n) option remember; `if`(n<4, [1, 2, 6, 20][n+1],
%p 2*a(n-1) +2*a(n-2) +5*a(n-3))
%p end:
%p seq(a(n), n=0..40);
%t Join[{1}, LinearRecurrence[{2, 2, 5}, {2, 6, 20}, 40]] (* _Jean-François Alcover_, May 29 2018 *)
%Y Even bisection of main diagonal of A247706.
%Y Cf. A174249, A233427, A247121.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Nov 17 2014
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