A250662 -id:A250662 - cp's OEIS frontend

A005178 Number of domino tilings of 4 X (n-1) board.

0, 1, 1, 5, 11, 36, 95, 281, 781, 2245, 6336, 18061, 51205, 145601, 413351, 1174500, 3335651, 9475901, 26915305, 76455961, 217172736, 616891945, 1752296281, 4977472781, 14138673395, 40161441636, 114079985111, 324048393905

Offset: 0

N. J. A. Sloane, David Singmaster, Frans J. Faase

Or, number of perfect matchings in graph P_4 X P_{n-1}.
a(0) = 0, a(1) = 1 by convention.
It is easy to see that the g.f. for indecomposable tilings, i.e., those that cannot be split vertically into smaller tilings, is g = x + 4x^2 + 2x^3 + 3x^4 + 2x^5 + 3x^6 + 2x^7 + 3x^8 + ... = x + 4x^2 + x^3*(2+3x)/(1-x^2); then g.f. = 1/(1-g) = (1-x^2)/(1-x-5x^2-x^3+x^4). - Emeric Deutsch, Oct 16 2006
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 22 2008
From Artur Jasinski, Dec 20 2008: (Start)
All numbers in this sequence are:
congruent to 0 mod 100 if n is congruent to 14 or 29 mod 30
congruent to 1 mod 100 if n is congruent to 0 or 1 or 12 or 16 or 27 or 28 mod 30
congruent to 5 mod 100 if n is congruent to 2 or 11 or 17 or 26 mod 30
congruent to 11 mod 100 if n is congruent to 3 or 25 mod 30
congruent to 36 mod 100 if n is congruent to 4 or 9 or 19 or 24 mod 30
congruent to 45 mod 100 if n is congruent to 8 or 20 mod 30
congruent to 51 mod 100 if n is congruent to 13 or 15 mod 30
congruent to 61 mod 100 if n is congruent to 10 or 18 mod 30
congruent to 81 mod 100 if n is congruent to 6 or 7 or 21 or 22 mod 30
congruent to 95 mod 100 if n is congruent to 5 or 23 mod 30
(End)
This is the case P1 = 1, P2 = -7, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

			For n=2 the graph is
  o-o-o-o
and there is one perfect tiling:
  o-o o-o
For n=3 the graph is
  o-o-o-o
  | | | |
  o-o-o-o
and there are five perfect tilings:
  o o o o
  | | | |
  o o o o
two like:
  o o o-o
  | | ...
  o o o-o
and this
  o-o o-o
  .......
  o-o o-o
and this
  o o-o o
  | ... |
  o o-o o
a(n+1)=r(n)-r(n-2), r(n)=if n=0 then 1 else sum(sum(binomial(k,j)*sum(binomial(j,i-j)*5^(i-j)*binomial(k-j,n-i-3*(k-j))*(-1)^(n-i-3*(k-j)),i,j,n-k+j),j,0,k),k,1,n), n>1. - _Vladimir Kruchinin_, Sep 08 2010

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 292.

T. D. Noe, Table of n, a(n) for n = 0..200
Steve Butler, Jason Ekstrand, and Steven Osborne, Counting Tilings by Taking Walks in a Graph, A Project-Based Guide to Undergraduate Research in Mathematics, Birkhäuser, Cham (2020), see page 158.
Curtis Cooper and Robert E. Kennedy, Problem B-735: Soln 1, Problem B-735: Soln 2, Square Root of a Recurrence, The Fibonacci Quarterly, 32(2):185-186, May 1994, and 32(4):374-375, Aug 1994.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Simone Rinaldi and D. G. Rogers, Indecomposability: polyominoes and polyomino tilings, The Mathematical Gazette 92.524 (2008): 193-204.
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Thotsaporn "Aek" Thanatipanonda, Statistics of Domino Tilings on a Rectangular Board, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Li Zhou, Northwestern University Math Problem Solving Group, Christopher Carl Heckman and Jerry Minkus, Tiling 4-Rowed Rectangles with Dominoes: 11187, The American Mathematical Monthly 114 (2007): 554-556.
Index to divisibility sequences
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (1,5,1,-1).

Row 4 of array A099390.
For all matchings see A033507.
Cf. A003775, A028468, A028469, A028470.
Cf. A003757. - T. D. Noe, Dec 22 2008
Bisection (odd part) gives A188899. - Alois P. Heinz, Oct 28 2012
Column k=2 of A250662.

a[0]:=1: a[1]:=1: a[2]:=5: a[3]:=11: for n from 4 to 26 do a[n]:=a[n-1]+5*a[n-2]+a[n-3]-a[n-4] od: seq(a[n],n=0..26); # Emeric Deutsch, Oct 16 2006
A005178:=-(-1-4*z-z**2+z**3)/(1-z-5*z**2-z**3+z**4) # conjectured (correctly) by Simon Plouffe in his 1992 dissertation; gives sequence apart from an initial 1

CoefficientList[Series[x(1-x^2)/(1-x-5x^2-x^3+x^4), {x,0,30}], x] (* T. D. Noe, Dec 22 2008 *)
LinearRecurrence[{1, 5, 1, -1}, {0, 1, 1, 5}, 28] (* Robert G. Wilson v, Aug 08 2011 *)
a[0] = 0; a[n_] := Product[2(2+Cos[2j Pi/5]+Cos[2k Pi/n]), {k, 1, (n-1)/2}, {j, 1, 2}] // Round;
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 20 2018 *)

r(n):=if n=0 then 1 else sum(sum(binomial(k,j)*sum(binomial(j,i-j)*5^(i-j)*binomial(k-j,n-i-3*(k-j))*(-1)^(n-i-3*(k-j)),i,j,n-k+j),j,0,k),k,1,n); a(n):=r(n)-r(n-2); /* Vladimir Kruchinin, Sep 08 2010 */

a(n) = a(n-1) + 5*a(n-2) + a(n-3) - a(n-4).
G.f.: x*(1 - x^2)/(1 - x - 5*x^2 - x^3 + x^4).
Limit_{n->oo} a(n)/a(n-1) = (1 + sqrt(29) + sqrt(14 + 2*sqrt(29)))/4 = 2.84053619409... - Philippe Deléham, Jun 12 2005
a(n) = (5*sqrt(29)/145)*(((1+sqrt(29)+sqrt(14+2*sqrt(29)))/4)^n+((1+sqrt(29)-sqrt(14+2*sqrt(29)))/4)^n-((1-sqrt(29)+sqrt(14-2*sqrt(29)))/4)^n-((1-sqrt(29)-sqrt(14-2*sqrt(29)))/4)^n). - Tim Monahan, Jul 30 2011
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(29))/4 and beta = (1 - sqrt(29))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 7/4; 1, 1/2].
a(n) = U(n-1,i*(1 + sqrt(5))/4)*U(n-1,i*(1 - sqrt(5))/4), where U(n,x) denotes the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = A129113(n+2) - A129113(n). - R. J. Mathar, May 03 2021

Amalgamated with (former) A003692, Dec 30 1995
Name changed and 0 prepended by T. D. Noe, Dec 22 2008
Edited by N. J. A. Sloane, Nov 15 2009

A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 19, 281, 1, 1, 1, 1, 1, 1, 57, 1183, 1, 1, 1, 1, 1, 1, 26, 121, 6728, 1, 1, 1, 1, 1, 1, 1, 75, 783, 31529, 1, 1, 1, 1, 1, 1, 1, 34, 154, 2861, 167089, 1, 1, 1, 1, 1, 1, 1, 1, 95, 269, 8133, 817991, 1, 1

Offset: 0

Alois P. Heinz, Nov 29 2014

A(n,n) = A034856(n+2) for n>=2.

			Square array A(n,k) begins:
  1, 1,      1,    1,    1,   1,   1,   1,  1, ...
  1, 1,      1,    1,    1,   1,   1,   1,  1, ...
  1, 1,     13,    1,    1,   1,   1,   1,  1, ...
  1, 1,     41,   19,    1,   1,   1,   1,  1, ...
  1, 1,    281,   57,   26,   1,   1,   1,  1, ...
  1, 1,   1183,  121,   75,  34,   1,   1,  1, ...
  1, 1,   6728,  783,  154,  95,  43,   1,  1, ...
  1, 1,  31529, 2861,  269, 190, 117,  53,  1, ...
  1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...

Alois P. Heinz, Antidiagonals n = 0..35, flattened
Wikipedia, Polyomino

Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.

Cf. A034856, A250662.

b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;
      if n=0 then 1
    elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
    else for k while l[k]>0 do od;
         `if`(n2*d+1 or max(l[k..k+d-1][])>0, 0,
          b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))
      fi
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, k}, Which[n == 0, 1,  Min[l] > 0, Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0 , k++]; If[n d]]] + If[d == 1 || k > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0,  0,  b[n, Join[l[[1 ;; k-1]], Array[1&, d],  l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A236582 The number of tilings of an 8 X n floor with 1 X 4 tetrominoes.

Original entry on oeis.org

1, 1, 1, 1, 7, 15, 25, 37, 100, 229, 454, 811, 1732, 3777, 7858, 15339, 31273, 65536, 136600, 276535, 562728, 1159942, 2400783, 4918159, 10052140, 20627526, 42480474, 87254743, 178855138, 366854368

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

R. J. Mathar, Paving rectangular regions..., arXiv:1311.6135 [math.CO], 2013, Table 37.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq. (28).

Cf. A003269 (4 X n floor), A236579 - A236581.
Column k=4 of A250662.
Cf. A251074.

p := (1-x)^3*(x+1)^3*(x^2+1)^3*(x^6-x^4-x^3-x^2+1) ;
q :=  -x^2 -13*x^10 -5*x^18 +8*x^6 -x -x^20 -9*x^4 +16*x^8 -13*x^12 -2*x^19 +1 +10*x^14 +5*x^7 +6*x^15 -6*x^11 +x^22 +6*x^16 +x^17 +2*x^5 -2*x^13 ;
taylor(p/q,x=0,30) ;
gfun[seriestolist](%) ;

G.f.: p(x)/q(x) with polynomials p and q defined in the Maple code.

A236577 The number of tilings of a 6 X n floor with 1 X 3 trominoes.

Original entry on oeis.org

1, 1, 1, 6, 13, 22, 64, 155, 321, 783, 1888, 4233, 9912, 23494, 54177, 126019, 295681, 687690, 1600185, 3738332, 8712992, 20293761, 47337405, 110368563, 257206012, 599684007, 1398149988, 3259051800, 7597720649, 17712981963

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 21.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq (14).
Index entries for linear recurrences with constant coefficients, signature (1,1,7,-1,-5,-10,-1,3,5,1,-1,-1).

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576 - A236578.
Column k=3 of A250662.
Cf. A251073.

g := (1-x^3)^2*(-x^2+1-x^3)/ (-x^10+x^12+x^11+10*x^6-5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

CoefficientList[Series[(1 - x^3)^2*(-x^2 + 1 - x^3)/(-x^10 + x^12 + x^11 + 10*x^6 - 5*x^9 - 3*x^8 + x^7 + x^4 - 7*x^3 + 5*x^5 - x^2 - x + 1), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)

x='x+O('x^50); Vec((1-x^3)^2*(-x^2+1-x^3)/(-x^10+x^12+x^11+10*x^6 -5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1)) \\ G. C. Greubel, Apr 27 2017

G.f.: See the definition of g in the Maple code.

A247117 Number of tilings of a 10 X n rectangle using 2n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 17, 28, 41, 56, 144, 317, 609, 1060, 1716, 3324, 6713, 13188, 24624, 43620, 80464, 153645, 296025, 562097, 1037921, 1920661, 3600832, 6820873, 12920804, 24211457, 45173688, 84493668, 158848825, 299451277, 562923960, 1055117520, 1976475968

Offset: 0

Alois P. Heinz, Nov 19 2014

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

Cf. A174249, A233427, A003520 (5 X n), A247218 (15 X n).

Column k=5 of A250662.

gf:= -(x^10+x^8-x^6-2*x^5-x^4-x^3+1) *(x-1)^4 *(x^4+x^3+x^2+x+1)^4 / (x^35 +x^33 -2*x^31 -7*x^30 -2*x^29 -6*x^28 +x^27 +9*x^26 +22*x^25 +8*x^24 +15*x^23 -4*x^22 -15*x^21 -39*x^20 -12*x^19 -20*x^18 +6*x^17 +10*x^16 +45*x^15 +8*x^14 +19*x^13 -4*x^12 -4*x^11 -33*x^10 -6*x^9 -10*x^8 +x^7 -3*x^6 +12*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

A250663 Number of tilings of a 12 X n rectangle using 2n hexominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 9, 19, 31, 45, 61, 79, 196, 419, 786, 1341, 2134, 3221, 5789, 10995, 20621, 37149, 63931, 105379, 180201, 319826, 578034, 1040971, 1840549, 3171726, 5465324, 9529019, 16830425, 29914626, 53016504, 92934619, 161999425, 282619059, 495436514

Offset: 0

Alois P. Heinz, Nov 26 2014

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

Column k=6 of A250662.

Cf. A251075.

gf:= -(x^15-x^12-2*x^10-2*x^9+x^7+2*x^6+x^5+x^4+x^3-1) *(x-1)^5 *(x+1)^5 *(x^2+x+1)^5 *(x^2-x+1)^5 / (x^51 -x^48 -3*x^46 -8*x^45 +2*x^43 +8*x^42 +3*x^41 +18*x^40 +28*x^39 -x^38 -11*x^37 -29*x^36 -15*x^35 -45*x^34
-56*x^33 +5*x^32 +24*x^31 +61*x^30 +30*x^29 +60*x^28 +70*x^27 -10*x^26 -25*x^25 -80*x^24 -30*x^23 -45*x^22 -61*x^21 +10*x^20 +10*x^19 +71*x^18 +15*x^17 +28*x^16 +38*x^15 -5*x^14 -2*x^13 -43*x^12 -8*x^11 -8*x^10 -13*x^9 +x^8 -4*x^7 +14*x^6 +x^3 +x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

G.f.: See Maple program.

A250664 Number of tilings of a 14 X n rectangle using 2n heptominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 10, 21, 34, 49, 66, 85, 106, 256, 535, 985, 1654, 2596, 3871, 5545, 9391, 16956, 30589, 53481, 89851, 145152, 226297, 364656, 610062, 1045297, 1799392, 3065145, 5121255, 8359876, 13624960, 22431292, 37434945, 63098713, 106641142, 179356873

Offset: 0

Alois P. Heinz, Nov 26 2014

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

Column k=7 of A250662.

Cf. A251076.

gf:= -(x^21 +x^18 -2*x^15 -3*x^14 -2*x^12 -2*x^11 +x^9 +2*x^8 +3*x^7 +x^6 +x^5 +x^4-1) *(x-1)^6 *(x^6+x^5+x^4+x^3+x^2+x+1)^6 / (x^70 +x^67 -3*x^64 -10*x^63 -3*x^61 -9*x^60 +3*x^58 +23*x^57 +45*x^56 +3*x^55 +21*x^54 +36*x^53 -x^52 -19*x^51 -76*x^50 -121*x^49 -18*x^48 -63*x^47 -84*x^46 +6*x^45 +51*x^44 +140*x^43 +216*x^42 +45*x^41 +105*x^40 +126*x^39
-15*x^38 -75*x^37 -154*x^36 -267*x^35 -60*x^34 -105*x^33 -126*x^32 +20*x^31 +65*x^30 +98*x^29 +236*x^28 +45*x^27 +63*x^26 +90*x^25 -15*x^24 -33*x^23 -40*x^22 -153*x^21 -18*x^20 -33*x^19 -48*x^18 +6*x^17 +15*x^16 +8*x^15 +69*x^14 +9*x^13 +9*x^12 +15*x^11 -x^10 -x^9 +5*x^8 -17*x^7 -x^4 -x +1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: See Maple program.

A250665 Number of tilings of a 16 X n rectangle using 2n octominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 11, 23, 37, 53, 71, 91, 113, 137, 324, 665, 1206, 1999, 3102, 4579, 6500, 8941, 14428, 24963, 43558, 74221, 122158, 193995, 298020, 444445, 682749, 1087886, 1781424, 2948681, 4861063, 7903814, 12610046, 19701987, 30702835, 48251196

Offset: 0

Alois P. Heinz, Nov 26 2014

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

Column k=8 of A250662.

Cf. A251077.

gf:= -(x^12 +x^11 +x^10 +2*x^9 +2*x^8 +2*x^7 +3*x^6 +2*x^5 +x^4 +2*x^3 +x^2 +1) *(x^15 -x^12 -x^11 -x^8 -2*x^7 +2*x^5 +3*x^4 +2*x^3 -x -1) *(x-1)^7 *(x+1)^7 *(x^2+1)^7 *(x^4+1)^7 / (x^91 +x^90 +x^89 +x^88 -4*x^84 -15*x^83 -15*x^82 -15*x^81 -12*x^80 -x^79 -x^78 +5*x^77 +41*x^76
+96*x^75 +96*x^74 +93*x^73 +67*x^72 +12*x^71 +8*x^70 -38*x^69 -182*x^68 -347*x^67 -346*x^66 -324*x^65 -225*x^64 -59*x^63 -31*x^62 +123*x^61 +459*x^60 +789*x^59 +782*x^58 +712*x^57 +496*x^56 +159*x^55 +75*x^54 -219*x^53 -723*x^52 -1185*x^51 -1164*x^50 -1038*x^49 -744*x^48 -261*x^47 -121*x^46 +229*x^45 +733*x^44 +1195*x^43 +1160*x^42 +1020*x^41 +768*x^40
+271*x^39 +131*x^38 -135*x^37 -471*x^36 -808*x^35 -773*x^34 -675*x^33 -549*x^32 -177*x^31 -93*x^30 +33*x^29 +198*x^28 +384*x^27 +363*x^26 +321*x^25 +283*x^24 +76*x^23 +48*x^22 -7*x^21 -71*x^20 -147*x^19 -140*x^18 -123*x^17 -118*x^16 -35*x^15 -24*x^14 -13*x^13 -2*x^12 +16*x^11 +15*x^10 +14*x^9 +20*x^8 +x^7 +x^6 +x^5 +x^4 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);

G.f.: See Maple program.

A250666 Number of tilings of a 18 X n rectangle using 2n nonominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 25, 40, 57, 76, 97, 120, 145, 172, 400, 809, 1449, 2376, 3652, 5345, 7529, 10284, 13696, 21232, 35417, 60028, 100004, 161664, 252945, 383660, 565776, 813712, 1201856, 1838369, 2895233, 4629793, 7412665, 11761912, 18384420

Offset: 0

Alois P. Heinz, Nov 26 2014

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

Column k=9 of A250662.

Cf. A251078.

gf:= -(x^36 +x^32 -3*x^28 -4*x^27 -3*x^24 -3*x^23 +3*x^20 +6*x^19 +6*x^18 +3*x^16 +4*x^15 +3*x^14 -x^12 -2*x^11 -3*x^10 -4*x^9 -x^8 -x^7 -x^6 -x^5 +1) *(x-1)^8 *(x^2+x +1)^8 *(x^6+x^3+1)^8 / (x^117 +x^113 -4*x^109 -13*x^108 -4*x^105 -12*x^104 +6*x^101 +43*x^100 +78*x^99 +6*x^97 +40*x^96 +66*x^95 -4*x^93 -55*x^92 -209*x^91 -286*x^90 -4*x^89 -52*x^88 -180*x^87 -220*x^86 +x^85 +33*x^84 +225*x^83 +605*x^82 +716*x^81
+32*x^80 +200*x^79 +480*x^78 +495*x^77 -8*x^76 -120*x^75 -540*x^74 -1155*x^73 -1295*x^72 -112*x^71 -448*x^70 -840*x^69 -792*x^68 +28*x^67 +252*x^66 +840*x^65 +1518*x^64 +1744*x^63 +224*x^62 +644*x^61 +1008*x^60 +924*x^59 -56*x^58 -336*x^57 -882*x^56 -1386*x^55 -1772*x^54 -280*x^53 -616*x^52 -840*x^51 -792*x^50 +70*x^49 +294*x^48 +630*x^47 +858*x^46
+1365*x^45 +224*x^44 +392*x^43 +480*x^42 +503*x^41 -56*x^40 -168*x^39 -300*x^38 -354*x^37 -803*x^36 -112*x^35 -160*x^34 -204*x^33 -244*x^32 +28*x^31 +60*x^30 +114*x^29 +103*x^28 +362*x^27 +32*x^26 +62*x^25 +72*x^24 +90*x^23 -8*x^22 -20*x^21 -31*x^20 -13*x^19 -118*x^18 -12*x^17 -12*x^16 -12*x^15 -20*x^14 +x^13 +x^12 +x^11 -7*x^10 +22*x^9 +x^5 +x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);

G.f.: See Maple program.

A250667 Number of tilings of a 20 X n rectangle using 2n decominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 27, 43, 61, 81, 103, 127, 153, 181, 211, 484, 967, 1714, 2785, 4246, 6169, 8632, 11719, 15520, 20131, 30169, 48753, 80533, 131499, 209215, 323073, 484567, 707587, 1008733, 1407649, 2011933, 2972524, 4525434, 7018281, 10944565

Offset: 0

Alois P. Heinz, Nov 26 2014

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

Column k=10 of A250662.

Cf. A251079.

gf:= -(x^45 -x^40 -4*x^36 -4*x^35 +3*x^31 +4*x^30 +6*x^27 +9*x^26 +6*x^25 -3*x^22 -6*x^21 -6*x^20 -4*x^18 -6*x^17 -6*x^16 -4*x^15 +x^13 +2*x^12 +3*x^11 +4*x^10 +x^9 +x^8 +x^7 +x^6 +x^5 -1) *(x-1)^9 *(x+1)^9 *(x^4+x^3+x^2+x+1)^9 *(x^4-x^3+x^2-x+1)^9 / (x^145 -x^140 -5*x^136 -14*x^135 +4*x^131 +14*x^130 +10*x^127 +60*x^126 +91*x^125 -6*x^122
-47*x^121 -91*x^120 -10*x^118 -105*x^117 -330*x^116 -364*x^115 +4*x^113 +61*x^112 +252*x^111 +364*x^110 +5*x^109 +95*x^108 +500*x^107 +1100*x^106 +1001*x^105 -x^104 -37*x^103 -280*x^102 -814*x^101 -1002*x^100 -45*x^99 -405*x^98 -1425*x^97 -2475*x^96 -2002*x^95 +9*x^94 +153*x^93 +765*x^92 +1760*x^91 +2011*x^90 +180*x^89 +1020*x^88
+2700*x^87 +3960*x^86 +3003*x^85 -36*x^84 -372*x^83 -1380*x^82 -2673*x^81 -3039*x^80 -420*x^79 -1680*x^78 -3570*x^77 -4620*x^76 -3432*x^75 +84*x^74 +588*x^73 +1722*x^72 +2904*x^71 +3516*x^70 +630*x^69 +1890*x^68 +3360*x^67 +3960*x^66 +3003*x^65 -126*x^64 -630*x^63 -1512*x^62 -2244*x^61 -3129*x^60 -630*x^59 -1470*x^58 -2250*x^57 -2475*x^56
-2011*x^55 +126*x^54 +462*x^53 +930*x^52 +1188*x^51 +2137*x^50 +420*x^49 +780*x^48 +1050*x^47 +1136*x^46 +1037*x^45 -84*x^44 -228*x^43 -390*x^42 -412*x^41 -1121*x^40 -180*x^39 -270*x^38 -379*x^37 -411*x^36 -418*x^35 +36*x^34 +72*x^33 +132*x^32 +98*x^31 +454*x^30 +45*x^29 +91*x^28
+114*x^27 +114*x^26 +127*x^25 -9*x^24 -22*x^23 -34*x^22 -9*x^21 -136*x^20 -14*x^19 -14*x^18 -14*x^17 -14*x^16 -23*x^15 +x^14 +x^13 +x^12 -8*x^11 +24*x^10 +x^5 +x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);

G.f.: See Maple program.

cp's OEIS Frontend

A005178 Number of domino tilings of 4 X (n-1) board.

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