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A281791 Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer. 1
3, 18, 10, 8, 18, 24, 32, 52, 68, 100, 142, 196, 280, 388, 542, 756, 1046, 1452, 2006, 2768, 3816, 5248, 7212, 9896, 13562, 18568, 25392, 34692, 47354, 64580, 88002, 119824, 163034, 221672, 301200, 409004, 555060, 752844, 1020550, 1382732, 1872520, 2534596, 3429206, 4637556, 6269070 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..44.

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-2,-1).

FORMULA

a(n) = S_1(2n+1) + S_5(2n+1) + S_3(2n+1) for n>1 where

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.

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.

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.

Where [m] is floor(m).

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

a(n) = 2*(A228577(n-1)+A228577(n+1))+4*(A182097(n-2)+A182097(n-1)), n>1. - R. J. Mathar, Apr 10 2017

EXAMPLE

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.

PROG

(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

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

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

a(n)=s1(n) + s3(n) + s5(n) \\ Charles R Greathouse IV, Feb 20 2017

CROSSREFS

Cf. A271786 [3X(2n+1) floor]. 2nd column of A272474.

Sequence in context: A324554 A007475 A324889 * A281722 A098874 A077104

Adjacent sequences:  A281788 A281789 A281790 * A281792 A281793 A281794

KEYWORD

nonn,easy,changed

AUTHOR

Yasutoshi Kohmoto, Jan 30 2017

STATUS

approved

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Last modified September 20 18:35 EDT 2019. Contains 327245 sequences. (Running on oeis4.)