|
%I #10 Aug 21 2024 09:58:54
%S 1,0,6,0,92,0,2376,5504,71248,253952,2175992,10140032,72393152,
%T 377186688,2462850896,13696293888,85330361248,489397951936,
%U 2980753350440,17385409283712,104607950578584,615347939509760,3679860151206528,21744078445007360,129606354640045704
%N Number of tilings of a 7 X n rectangle using right trominoes and 2 X 2 tiles.
%H Alois P. Heinz, <a href="/A219948/b219948.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_55">Index entries for linear recurrences with constant coefficients</a>, signature (4, 23, -80, 115, -454, -448, 14170, -29225, 7046, -146051, 588502, 846767, -3992062, -2604251, 5563402, 51185474, -68603352, -54073841, -158364404, 578867289, -174549722, 760215019, -1876828610, 149204601, -7282850020, 9226439666, 5199134488, 26926334975, -31896621428, -18394521038, -82151994152, 28473878368, -18476692408, 93062076382, -574037808, 70728329804, -90109498688, 37654878912, -94984995808, 44055137768, -44217857024, 44741808416, 3157624320, 7979568544, -1430120704, -6084576320, 6323347968, -1759063552, -178826240, 223708160, -307560448, 79364096, -26296320, 8060928, -393216).
%F G.f.: see Maple program.
%e a(2) = 6, because there are 6 tilings of a 7 X 2 rectangle using right trominoes and 2 X 2 tiles:
%e .___. .___. .___. .___. .___. .___.
%e | . | | . | | ._| | . | | . | |_. |
%e |___| |___| |_| | |___| |___| | |_|
%e | . | | ._| |___| | . | |_. | |___|
%e |___| |_| | | . | |___| | |_| | . |
%e | ._| |___| |___| |_. | |___| |___|
%e |_| | | . | | . | | |_| | . | | . |
%e |___| |___| |___| |___| |___| |___|
%p gf:= -(65536*x^53 -1146880*x^52 +1056768*x^51 -11173888*x^50 -509952*x^49 +33372672*x^48 -46419968*x^47 +225738880*x^46 -47477504*x^45 -389283328*x^44 +1604376704*x^43 -193165168*x^42 +1816657344*x^41 +843167448*x^40 -5731518112*x^39 +3110216464*x^38 -11976005632*x^37 +6543273808*x^36 -5367184032*x^35 +9797996038*x^34 +2933363944*x^33 +6282799800*x^32 -2982763584*x^31 +85793812*x^30 -8254807988*x^29 +58758079*x^28 -1538296008*x^27 +1483118884*x^26 +420789512*x^25 +260010263*x^24 -408844686*x^23 +66645661*x^22 -78341234*x^21 -34068549*x^20 +45496788*x^19 +25092255*x^18 -16579172*x^17 -6253012*x^16 +4608446*x^15 +354299*x^14 -225506*x^13 -189351*x^12 +193666*x^11 -51177*x^10 +17850*x^9 -15917*x^8 +8086*x^7 -18*x^6 -566*x^5 +161*x^4 -56*x^3 +17*x^2 +4*x -1) /
%p (393216*x^55 -8060928*x^54 +26296320*x^53 -79364096*x^52 +307560448*x^51 -223708160*x^50 +178826240*x^49 +1759063552*x^48 -6323347968*x^47 +6084576320*x^46 +1430120704*x^45 -7979568544*x^44 -3157624320*x^43 -44741808416*x^42 +44217857024*x^41 -44055137768*x^40 +94984995808*x^39 -37654878912*x^38 +90109498688*x^37 -70728329804*x^36 +574037808*x^35 -93062076382*x^34 +18476692408*x^33 -28473878368*x^32 +82151994152*x^31 +18394521038*x^30 +31896621428*x^29 -26926334975*x^28 -5199134488*x^27 -9226439666*x^26 +7282850020*x^25 -149204601*x^24 +1876828610*x^23 -760215019*x^22 +174549722*x^21 -578867289*x^20 +158364404*x^19 +54073841*x^18 +68603352*x^17 -51185474*x^16 -5563402*x^15 +2604251*x^14 +3992062*x^13 -846767*x^12 -588502*x^11 +146051*x^10 -7046*x^9 +29225*x^8 -14170*x^7 +448*x^6 +454*x^5 -115*x^4 +80*x^3 -23*x^2 -4*x +1):
%p a:= n-> coeff (series (gf, x, n+1), x, n):
%p seq(a(n), n=0..30);
%Y Column k=7 of A219946.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 01 2012
|
