A049086 -id:A049086

Search: a049086 -id:a049086

Displaying 1-5 of 5 results found.

page 1

A046984

Number of ways to tile a 4 X 3n rectangle with right trominoes.

+10
6

1, 4, 18, 88, 468, 2672, 16072, 100064, 636368, 4097984, 26579488, 173093760, 1129796928, 7383588608, 48287978624, 315921649152, 2067346607360, 13530037877760, 88555066819072, 579620448450560, 3793872862974976, 24832858496561152, 162544900186359808

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The sequence of tiling 2 X 3n rectangles with L-trominoes is 2^n. The sequence of tiling 3 X 2n rectangles is 2^n. All these tilings have vertical faults but no horizontal faults. - R. J. Mathar, Dec 08 2022

This sequence is the Hadamard sum of the following 4 sequences: 0, 0, 16, 64, 256, 1024, 4096... (A000302, tilings which have both vertical and horizontal faults), 0, 4, 0, 0, 0, 0, 0, ...(tilings which have horizontal but no vertical faults), 0, 0, 0, 16, 164, 1360, 10248, 73312, 508624, 3462592, 23291424.. (tilings which have vertical but no horizontal faults), 1, 0, 2, 8, 48, 288, 1728, 10368,.. (essentially A084477, tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

REFERENCES

Suggested on p. 96 of 1994 edition of "Polyominoes" by Samuel W. Golomb.

LINKS

Table of n, a(n) for n=0..22.

R. J. Mathar, Fault-free tilings with dominoes or trominoes.

Cristopher Moore, Preprint and figures

Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.

Index entries for linear recurrences with constant coefficients, signature (10,-22,-4).

FORMULA

G.f.: (1 - 6*x)/(1 - 10*x + 22*x^2 + 4*x^3).

a(0)=1, a(1)=4, a(2)=18, a(n)=10*a(n-1)-22*a(n-2)-4*a(n-3). - Harvey P. Dale, Mar 31 2012

MAPLE

a:= n-> (<<0|1|0>, <0|0|1>, <-4|-22|10>>^n. <<1, 4, 18>>)[1, 1]:

seq(a(n), n=0..22); # Alois P. Heinz, Feb 21 2022

MATHEMATICA

CoefficientList[Series[(1-6x)/(1-10x+22x^2+4x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{10, -22, -4}, {1, 4, 18}, 40] (* Harvey P. Dale, Mar 31 2012 *)

PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; -4, -22, 10]^n*[1; 4; 18])[1, 1] \\ Charles R Greathouse IV, Feb 10 2017

CROSSREFS

Cf. A084478 (5 X 3n), A351323 (6 X n), A351324 (7 X 3n), A049086 (straight trominoes), A233339 (mixed trominoes).

KEYWORD

nonn,easy,nice

AUTHOR

Cristopher Moore (moore(AT)santafe.edu)

STATUS

approved

A236576

The number of tilings of a 5 X (3n) floor with 1 X 3 trominoes.

+10
5

1, 4, 22, 121, 664, 3643, 19987, 109657, 601624, 3300760, 18109345, 99355414, 545105209, 2990674357, 16408085929, 90021597712, 493896002842, 2709719309845, 14866649448256, 81564634762843, 447497579542135

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.

R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 20.

R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014.

Index entries for linear recurrences with constant coefficients, signature (6,-3,1).

FORMULA

G.f.: (1-x)^2/(1-6*x+3*x^2-x^3).

a(n) = 6*a(n-1) - 3*a(n-2) + a(n-3). - M. Poyraz Torcuk, Oct 24 2021

MAPLE

g := (1-x)^2/(1-6*x+3*x^2-x^3) ;

taylor(%, x=0, 30) ;

gfun[seriestolist](%) ;

MATHEMATICA

CoefficientList[Series[(1 - x)^2/(1 - 6 x + 3 x^2 - x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 29 2017 *)

LinearRecurrence[{6, -3, 1}, {1, 4, 22}, 30] (* M. Poyraz Torcuk, Nov 06 2021 *)

PROG

(PARI) my(x='x+O('x^50)); Vec((1-x)^2/(1-6*x+3*x^2-x^3)) \\ G. C. Greubel, Apr 29 2017

CROSSREFS

Cf. A000930 (3 X n floor), A049086 (4 X 3n floor), A236577, A236578.

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Jan 29 2014

STATUS

approved

A236577

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

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).

FORMULA

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

MAPLE

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](%) ;

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576 - A236578.

Column k=3 of A250662.

Cf. A251073.

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Jan 29 2014

STATUS

approved

A109955

Number triangle binomial(n+2k,3k).

+10
4

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 28, 10, 1, 1, 35, 84, 55, 13, 1, 1, 56, 210, 220, 91, 16, 1, 1, 84, 462, 715, 455, 136, 19, 1, 1, 120, 924, 2002, 1820, 816, 190, 22, 1, 1, 165, 1716, 5005, 6188, 3876, 1330, 253, 25, 1, 1, 220, 3003, 11440, 18564, 15504, 7315, 2024

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Riordan array (1/(1-x),x/(1-x)^3).

Inverse array is A109956. Row sums are A052544.

Diagonal sums are A034943(n+1).

LINKS

Table of n, a(n) for n=0..62.

FORMULA

Number triangle T(n, k) = binomial(n+2k, 3k).

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k) + T(n-1,k-1). - Philippe Deléham, Feb 18 2012

G.f.: (1-x)^2/((1-x)^3-y*x). - Philippe Deléham, Feb 18 2012

Sum_{k, 0<=k<=n} T(n,k)*x^k = A185963(n), A000012(n), A052544(n), A049086(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Feb 18 2012

EXAMPLE

Rows begin

1,1;

1,4,1;

1,10,7,1;

1,20,28,10,1;

1,35,84,55,13,1;

PROG

(PARI) tabl(nn) = {my(m = matrix(nn, nn, n, k, if (n<k, 0, binomial(n+2*k-3, 3*k-3)))); for (n=1, nn, for (k=1, n, print1(m[n, k], ", "); ); print(); ); } \\ Michel Marcus, Nov 20 2015

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jul 06 2005

STATUS

approved

A236578

The number of tilings of a 7 X (3n) floor with 1 X 3 trominoes.

+10
4

1, 9, 155, 2861, 52817, 972557, 17892281, 329097125, 6052932495, 111328274273, 2047599783121, 37660384283749, 692666924307063, 12739845501187821, 234317040993180833, 4309665744385061493, 79265335342431559977

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..785

R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 22.

R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq. (15).

FORMULA

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

MAPLE

p := (x-1)^2*(-x^15 +14*x^14 -104*x^13 +527*x^12 -1971*x^11 +5573*x^10 -11973*x^9 +19465*x^8 -23695*x^7 +21166*x^6 -13512*x^5 +5915*x^4 -1685*x^3 +291*x^2 -27*x+1) ;

q := -17*x^17 +293180*x^8 -236178*x^7 +142400*x^6 -62621*x^5 +19420*x^4 -4062*x^3 +533*x^2 -38*x +x^18 +1 +151*x^16 -946*x^15 +4558*x^14 -17135*x^13 +50164*x^12 -114198*x^11 +202080*x^10 -277277*x^9 ;

taylor(p/q, x=0, 30) ;

gfun[seriestolist](%) ;

CROSSREFS

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576, A236577.

KEYWORD

easy,nonn

AUTHOR

R. J. Mathar, Jan 29 2014

STATUS

approved

page 1

Search completed in 0.013 seconds