A273474 -id:A273474 - cp's OEIS frontend

A233289 Number of tilings of a 3 X 3 X n box using 3n bricks of shape 3 X 1 X 1.

1, 2, 4, 21, 92, 320, 1213, 4822, 18556, 70929, 273808, 1057020, 4069737, 15676666, 60424640, 232846801, 897164316, 3457096532, 13321674833, 51332757274, 197801848744, 762200458321, 2937024077340, 11317358546188, 43609682555721, 168043191679374

Offset: 0

Alois P. Heinz, Dec 06 2013

This is a variant of the Jenga game (see link).

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], 2014; eq. (40).
Wikipedia, Jenga
Index entries for linear recurrences with constant coefficients, signature (3,0,13,2,-11,-7,4,-3,1,-1)

Column k=3 of A233308.

Cf. A006253, A233291, A233294, A273474.

gf:= (x^7-x^6+x^5-x^4+4*x^3+2*x^2+x-1)/(-x^10+x^9
     -3*x^8+4*x^7-7*x^6-11*x^5+2*x^4+13*x^3+3*x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: (x^7 -x^6 +x^5 -x^4 +4*x^3 +2*x^2 +x -1) / (-x^10 +x^9 -3*x^8 +4*x^7 -7*x^6 -11*x^5 +2*x^4 +13*x^3 +3*x -1).

A237355 The number of tilings of the 3 X 4 X n room by 1 X 1 X 3 boxes.

Original entry on oeis.org

1, 3, 9, 92, 749, 4430, 30076, 217579, 1479055, 10046609, 69575902, 479035195, 3284657308, 22593041544, 155444686265, 1068352050847, 7344626541715, 50504764148658, 347234420131143, 2387280989007848, 16413850076764282, 112852648679477233, 775901649656851817

Offset: 0

R. J. Mathar, Feb 07 2014

nonn

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.

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).
Index entries for linear recurrences with constant coefficients, signature (5, -1, 176, -323, -293, -10064, 1074, 7860, 272389, 183389, -59839, -4323330, -4517295, -105217, 44954256, 50928278, 5385773, -322607030, -339177827, -33326860, 1656296393, 1420714989, 41410295, -6265092815, -3679914785, 385369833, 17881094620, 4934247701, -1839198593, -39629953212, 2139678101, 2160878871, 70543105470, -27232379880, 6729477185, -104511366128, 71881449601, -32366552001, 132185701967, -119917584642, 67890453774, -142211822794, 144862794083, -91273082294, 126560496081, -132197432121, 86012823396, -90553404712, 92190746241, -59232895357, 51135748066, -49378215194, 30858791790, -22568613842, 20609808196, -12491943945, 7780358999, -6816233581, 3979624295, -2091442392, 1795023861, -997380768, 436930703, -375021274, 196601781, -73736037, 60815785, -31180618, 10961366, -7072305, 4209081, -1578469, 559851, -506788, 211745, -55563, 36708, -12778, 12700, -1278, 130, -1761, -85, 65, 65, -33, 11, 7, 0, -1).

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

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) ;

cp's OEIS Frontend

A233289 Number of tilings of a 3 X 3 X n box using 3n bricks of shape 3 X 1 X 1.

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