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%I #35 Mar 01 2024 11:13:03
%S 0,1,2,10,36,144,556,2172,8452,32932,128260,499604,1945988,7579860,
%T 29524324,115000436,447938884,1744769748,6796063908,26471392948,
%U 103108894980,401620128916,1564353180772,6093322268020,23734139269316,92447000518484,360090914096676
%N a(n) = 3*a(n-1) + 4*a(n-2) - 2*a(n-3) with a(0)=0, a(1)=1, a(2)=2.
%C For n > 0, a(n) is the number of ways to tile a 2 X 2 X (n-1) box with 1 X 1 X 1 cubes and 1 X 2 X 2 plates.
%H Qianyu Guo, <a href="https://pioneeracademics.com/wp-content/plugins/wonderplugin-pdf-embed/pdfjslight/web/viewer.html?v=2&disabledownload=1&disableprint=1&disabletext=1&disabledoc=1&disableopenfile=1&disablerightclick=1&file=https%3A%2F%2Fpioneeracademics.com%2Fwp-content%2Fuploads%2F2022%2F03%2F2020-Pioneer-Research-Journal.pdf#%5B%7B%22num%22%3A762%2C%22gen%22%3A0%7D%2C%7B%22name%22%3A%22FitH%22%7D%2C698%5D">Cube-plate Tiling Numbers and Their Identities</a>, Pioneer Academics, Vol 7, pp. 227-237, 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,4,-2).
%F G.f.: (1 - x) / (1 - 3*x - 4*x^2 + 2*x^3). - _Colin Barker_, Jun 14 2020
%e Here are four of the a(4) = 36 possible tilings of a 2 x 2 x 3 box with cubes and plates:
%e . ______ ______ ______ _______
%e ./ / / /| / /___/| /___/ /| / / /|
%e /_/_/_/ | /_/___/|| /___/_/ | /_/___ //|
%e | | | | / | | ||/ | | | / | |___|//
%e |_|_|_|/ |_|___|/ |_ _|_|/ |_|___|/
%t LinearRecurrence[{3, 4, -2}, {0, 1, 2}, 30] (* _Greg Dresden_, Jun 14 2020 *)
%o (PARI) Vec((1 - x) / (1 - 3*x - 4*x^2 + 2*x^3) + O(x^30)) \\ _Colin Barker_, Jun 14 2020
%Y Cf. A006253, A237358, A237355, A237356.
%K nonn,easy
%O 0,3
%A _Qianyu Guo_, Jun 14 2020
%E More terms from _Colin Barker_, Jun 14 2020
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