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%I #32 Sep 07 2019 06:05:09
%S 3,18,10,8,18,24,32,52,68,100,142,196,280,388,542,756,1046,1452,2006,
%T 2768,3816,5248,7212,9896,13562,18568,25392,34692,47354,64580,88002,
%U 119824,163034,221672,301200,409004,555060,752844,1020550,1382732,1872520,2534596,3429206,4637556,6269070
%N Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer.
%C Apart from a single 1 X 1 monomer, the area is tiled with 2 X 1 mats. No four mats are permitted to meet at a point.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,-2,-1).
%F a(n) = S_1(2n+1) + S_5(2n+1) + S_3(2n+1) for n>1 where
%F S_1(n) = 2* Sum_{k= 0<=k<=[(n-1)/6]} ((n+3)/4-1/2*k) *((n-1)/4-1/2*k)!/(k!*((n-1)/4-3/2*k)!). The sum is over even k if n==1 (mod 4), else over odd k.
%F S_5(n) = 2* Sum_{0<=k<=[(n-5)/6]} ((n+7)/4-1/2*k) *((n-5)/4-1/2*k)!/(k!*((n-5)/4-3/2*k)!). The sum is over even k if n==1 (mod 4) else over odd k.
%F S_3(n) = 2* Sum_{0<=k<=[(n-3)/6]} 2*((n-3)/4-1/2*k)!/(k!*((n-3)/4-3/2*k)!). The sum is over odd k if n==1 (mod 4), else over even k.
%F Where [m] is floor(m).
%F G.f. x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. (Includes zeros for even floor widths).- _R. J. Mathar_, Apr 10 2017
%F a(n) = 2*(A228577(n-1)+A228577(n+1))+4*(A182097(n-2)+A182097(n-1)), n>1. - _R. J. Mathar_, Apr 10 2017
%e For n=0, the 5X1 floor allows the monomer to be placed at one of the two ends or in the middle: a(n=0)=3.
%o (PARI) s1(n)=my(s); forstep(k=(n%4!=1),(n-1)\6,2, s+=((n+3)/4-k/2)*((n-1)/4-k/2)!/(k!*((n-1)/4-3*k/2)!)); 2*s
%o s3(n)=my(s); forstep(k=(n%4==1),(n-3)\6,2, s+=((n-3)/4-k/2)!/(k!*((n-3)/4-3*k/2)!)); 2*s
%o s5(n)=my(s); forstep(k=(n%4!=1),(n-5)\6,2, s+=((n+7)/4-k/2)*((n-5)/4-k/2)!/(k!*((n-5)/4-3*k/2)!)); 2*s
%o a(n)=s1(n) + s3(n) + s5(n) \\ _Charles R Greathouse IV_, Feb 20 2017
%Y Cf. A271786 [3X(2n+1) floor]. 2nd column of A272474.
%K nonn,easy
%O 0,1
%A _Yasutoshi Kohmoto_, Jan 30 2017
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