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A237355 The number of tilings of the 3 X 4 X n room by 1 X 1 X 3 boxes. 2
1, 3, 9, 92, 749, 4430, 30076, 217579, 1479055, 10046609, 69575902, 479035195, 3284657308, 22593041544, 155444686265, 1068352050847, 7344626541715, 50504764148658, 347234420131143, 2387280989007848, 16413850076764282, 112852648679477233, 775901649656851817 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The count compiles all arrangements without respect to symmetry: Stacks that are equivalent after rotations or flips through any of the 3 axes or 3 planes are counted with multiplicity.

The rational generating function p(x)/q(x) is qualified in the Maple section for argument x=z^4.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

R. J. Mathar, Tilings of rectangular regions by rectangular tiles: counts derived from transfer matrices, arXiv:1406.7788 [math.CO], eq. (41).

MAPLE

A237355 := proc(n)

p :=

-(-1 +3396185*z^268 +126*z^12 -1276*z^28 +5*z^8 +

5283231061*z^124-1577588*z^52 -13645425693*z^152 +

144704015*z^244 -455*z^20 +13238456*z^260 +31819046103*z

^156 -z^320 +25108196325*z^148 -93044*z^44 -2901338989*z

^128 -10*z^336 -2163597596*z^216 +10067*z^32 -29481842170

*z^168 -27774785437*z^144 +2*z^4 -4*z^340 -18745905736

*z^176 +7299716699*z^140 +21*z^332 -501*z^312 +

32101048863*z^172 -71881307*z^76 +153863*z^36 +13993826*z

^64 +35536*z^292 +10372*z^300 -21028*z^296 -28*z^324

+21886734767*z^180 +2407716*z^68 +91622*z^40 -23376175*z

^80 -24005339291*z^184 +100677112*z^92 -7169*z^304 +1184*

z^316 +222741*z^56 -192795426*z^100 +5270172488*z^108 -

57*z^328 +25361870*z^252 -4094967*z^264 +943065389*z^

116 +19724145370*z^132 -32733186254*z^160 +18663190295*z^

164 +561417344*z^84 -1224595901*z^224 +14015301065*z^188 -

2024669*z^272 -11369197887*z^120 +337006779*z^236 -

990719131*z^112 -25766687*z^256 -6333*z^24 +207060338*z^

88 +19722918*z^60 -123289650*z^72 -73800398*z^248 +651316

*z^276 -309189319*z^104 -93*z^16 -14025522915*z^136 -

2171925*z^48 +13808940292*z^196 +671*z^308 +2217864262*z

^220 -634809849*z^232 +617029016*z^228 +259296*z^284 -

7967920024*z^200 -6205639852*z^208 +3519281640*z^212 +

5961180966*z^204 +z^348 -12884456696*z^192 -1943914891*z^

96 -105001*z^288 -138606683*z^240 -319172*z^280) ;

p := algsubs(z^4=x, p) ;

q :=

1-

60815785*z^268 -176*z^12 -1074*z^28 +z^8 -2139678101*z

^124 +4517295*z^52 +32366552001*z^152 -1795023861*z^244 +

293*z^20 -196601781*z^260 -132185701967*z^156 +1278*z^

320 -71881449601*z^148 +59839*z^44 -2160878871*z^128 -65*

z^336 +22568613842*z^216 -7860*z^32 +142211822794*z^168 +

104511366128*z^144 -5*z^4 -65*z^340 +91273082294*z^176

-6729477185*z^140 +85*z^332 +12778*z^312 -144862794083*z

^172 +339177827*z^76 -272389*z^36 -50928278*z^64 -559851*

z^292 -211745*z^300 +506788*z^296 -130*z^324 -

126560496081*z^180 -5385773*z^68 -183389*z^40 +33326860*z

^80 +132197432121*z^184 -41410295*z^92 +55563*z^304 -

12700*z^316 +105217*z^56 +3679914785*z^100 -17881094620*z

^108 +1761*z^328 -436930703*z^252 +z^360 +73736037*z^

264 +1839198593*z^116 -70543105470*z^132 +119917584642*z^

160 -67890453774*z^164 -1656296393*z^84 +12491943945*z^224

-86012823396*z^188 +31180618*z^272 -7*z^352 +39629953212*

z^120 -3979624295*z^236 -4934247701*z^112 +375021274*z^

256 +10064*z^24 -1420714989*z^88 -44954256*z^60 +322607030

*z^72 +997380768*z^248 -10961366*z^276 -385369833*z^104

+323*z^16 +27232379880*z^136 +4323330*z^48 -92190746241*

z^196 -36708*z^308 -20609808196*z^220 +6816233581*z^232 -

7780358999*z^228 -4209081*z^284 +59232895357*z^200 +

49378215194*z^208 -30858791790*z^212 -51135748066*z^204 -11

*z^348 +90553404712*z^192 +6265092815*z^96 +1578469*z^

288 +33*z^344 +2091442392*z^240 +7072305*z^280 ;

q := algsubs(z^4=x, q) ;

coeftayl(p/q, x=0, n) ;

end proc:

seq(A237355(n), n=0..20) ;

CROSSREFS

Cf. A233247 (2 X 3 X n rooms), A233289 (3 X 3 X n rooms), A273474.

Sequence in context: A018654 A003225 A203104 * A007663 A231212 A185174

Adjacent sequences:  A237352 A237353 A237354 * A237356 A237357 A237358

KEYWORD

nonn

AUTHOR

R. J. Mathar, Feb 07 2014

STATUS

approved

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Last modified November 15 13:20 EST 2019. Contains 329149 sequences. (Running on oeis4.)