A233289 -id:A233289 - cp's OEIS frontend

A233308 Number A(n,k) of tilings of a k X k X n box using k*n bricks of shape k X 1 X 1; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 2, 9, 1, 1, 2, 4, 32, 1, 1, 2, 4, 21, 121, 1, 1, 2, 4, 8, 92, 450, 1, 1, 2, 4, 8, 45, 320, 1681, 1, 1, 2, 4, 8, 16, 248, 1213, 6272, 1, 1, 2, 4, 8, 16, 93, 1032, 4822, 23409, 1, 1, 2, 4, 8, 16, 32, 668, 3524, 18556, 87362, 1, 1, 2, 4, 8, 16, 32, 189, 3440, 13173, 70929, 326041, 1

Offset: 0

Alois P. Heinz, Dec 07 2013

			Square array A(n,k) begins:
  1,     1,     1,     1,     1,     1, ...
  1,     2,     2,     2,     2,     2, ...
  1,     9,     4,     4,     4,     4, ...
  1,    32,    21,     8,     8,     8, ...
  1,   121,    92,    45,    16,    16, ...
  1,   450,   320,   248,    93,    32, ...
  1,  1681,  1213,  1032,   668,   189, ...
  1,  6272,  4822,  3524,  3440,  1832, ...
  1, 23409, 18556, 13173, 13728, 11976, ...

Alois P. Heinz, Antidiagonals n = 0..17, flattened

Columns k=1-6 give: A000012, A006253, A233289, A233291, A233294, A233424.

Diagonals include: A000079, A068156.

b:= proc(n, l) option remember; local d, t, k; d:= isqrt(nops(l));
      if max(l[])>n then 0 elif n=0 then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(x->x-t, l))
    else for k while l[k]>0 do od; b(n, subsop(k=d, l))+
         `if`(irem(k,d)=1 and {seq(l[k+j], j=1..d-1)}={0},
         b(n, [seq(`if`(h-k=0, 1, l[h]), h=1..nops(l))]), 0)+
         `if`(k<=d and {seq(l[k+d*j], j=1..d-1)}={0},
         b(n, [seq(`if`(irem(h-k, d)=0, 1, l[h]), h=1..nops(l))]), 0)
      fi
    end:
A:= (n, k)-> `if`(k>n, 2^n, b(n, [0$k^2])):
seq(seq(A(n, 1+d-n), n=0..d), d=0..11);

b[n_, l_] := b[n, l] = Module[{d, t, k}, d= Sqrt[Length[l]]; Which[ Max[l]>n, 0, n==0, 1, Min[l]>0, t=Min[l]; b[n-t, l-t], True, k=Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k->d]]+ If[Mod[k, d]==1 && Union[ Table[ l[[k+j]], {j, 1, d-1}]] == {0}, b[n, Table[ If [h-k=0, 1, l[[h]] ], {h, 1, Length[l]}]], 0]+ If[k <= d && Union[ Table[ l[[k+d*j]], {j, 1, d-1}]] == {0}, b[n, Table[ If[ Mod[h-k, d] == 0, 1, l[[h]] ], {h, 1, Length[l]}]], 0] ] ]; a[n_, k_]:= If[k>n, 2^n, b[n, Array[0&, k^2]]]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A(n,k) = 2^n = A000079(n) for k>n.

A(n,n) = A068156(n) for n>1.

A233291 Number of tilings of a 4 X 4 X n box using 4n bricks of shape 4 X 1 X 1.

Original entry on oeis.org

1, 2, 4, 8, 45, 248, 1032, 3524, 13173, 54274, 228712, 917992, 3608665, 14286188, 57438652, 231343468, 926921081, 3700936774, 14793198332, 59241396140, 237333611629, 950127617692, 3801974385964, 15215432779936, 60907523900693, 243826775063490, 976008753961184

Offset: 0

Alois P. Heinz, Dec 06 2013

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. (52).

Column k=4 of A233308.

Cf. A006253, A233289, A233294.

G.f.: (x^31 -x^30 -x^29 -3*x^28 -5*x^27 +8*x^26 +3*x^25 -39*x^24 +48*x^23 +34*x^22 -11*x^21 +112*x^20 -63*x^19 -34*x^18 +166*x^17 -188*x^16 -174*x^15 +151*x^14 -251*x^13 +57*x^12 +104*x^11 +171*x^10 -69*x^9 -90*x^8 -30*x^7 -61*x^6 +20*x^5 +31*x^4 +4*x^3 +4*x^2 +x -1) / (-x^35 +x^34 +x^33 +5*x^32 +4*x^31 -11*x^30 -10*x^29 +54*x^28 -65*x^27 -48*x^26 -48*x^25 -195*x^24 +231*x^23 -78*x^22 -473*x^21 +794*x^20 +447*x^19 -981*x^18 +1285*x^17 +395*x^16 -720*x^15 -202*x^14 +640*x^13 +567*x^12 -232*x^11 +638*x^10 -169*x^9 -233*x^8 -256*x^7 -97*x^6 +29*x^5 +52*x^4 -4*x^3 +2*x^2 +3*x -1).

A233294 Number of tilings of a 5 X 5 X n box using 5n bricks of shape 5 X 1 X 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 93, 668, 3440, 13728, 46252, 172577, 739002, 3417464, 15550336, 65956764, 271247405, 1119519052, 4726308568, 20348072952, 87598217268, 373660404281, 1581318625634, 6680851858676, 28326586367464, 120536842633616, 513461699313993

Offset: 0

Alois P. Heinz, Dec 06 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, G.f. for A233294

Column k=5 of A233308.

Cf. A006253, A233289, A233291.

G.f.: see link above.

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

A233308 Number A(n,k) of tilings of a k X k X n box using k*n bricks of shape k X 1 X 1; square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A233291 Number of tilings of a 4 X 4 X n box using 4n bricks of shape 4 X 1 X 1.

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A233294 Number of tilings of a 5 X 5 X n box using 5n bricks of shape 5 X 1 X 1.

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

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A273474 Number of tilings of a 3 X 5 X n box using 5n bricks of shape 3 X 1 X 1.

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