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Search: a234312 -id:a234312
Displaying 1-9 of 9 results found.
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A234931
Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, N.
+10
11
1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 8, 0, 16, 0, 40, 0, 64, 16, 200, 96, 504, 464, 1528, 1664, 4376, 5616, 12792, 18192, 38264, 58384, 115832, 186368, 355808, 589344, 1095408, 1853664, 3383656, 5802016, 10470376, 18125280, 32461312, 56552736, 100782696, 176318464
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
FORMULA
G.f.: (4*x^20 +4*x^18 +8*x^16 -3*x^14 +4*x^13 -5*x^12 -2*x^11 +3*x^10 -2*x^9 +6*x^8 -2*x^7 +2*x^6 -2*x^5 -x^4 +2*x -1) / (-8*x^22 -28*x^20 -6*x^18 +8*x^17 +26*x^16 +4*x^15 +7*x^14 -8*x^13 -9*x^12 -14*x^11 +7*x^10 +2*x^9 +8*x^8 -2*x^7 +2*x^6 -6*x^5 +x^4 +2*x -1).
EXAMPLE
a(4) = 2:
._______. ._______.
| | ._. | | ._. | |
| |_| |_| |_| |_| |
|_. |_. | | ._| ._|
| |_| | | | | |_| |
|_____|_| |_|_____|.
CROSSREFS
Cf. A174249, A233427, A234312, A247680, A247126.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 01 2014
STATUS
approved
A247443
Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, L, Y.
+10
10
1, 0, 2, 0, 6, 16, 32, 104, 186, 800, 1700, 4836, 11186, 29940, 84388, 208808, 563364, 1391664, 3787510, 9824684, 25712276, 66815444, 173151378, 457266220, 1188536784, 3113743272, 8087358736, 21152284376, 55283003950, 144314582896, 376852311434, 982507243820
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, G.f. and Maple program for A247443
Wikipedia, Pentomino
FORMULA
G.f.: see link.
EXAMPLE
a(5) = 16:
._._______. ._______._.
| | ._____| |_. .___| |
| |_| ._| | | |_| ._| |
| |_. |_. | | |_. |_. |
|___|_| | | | ._|_| |_|
|_______|_| (*8) |_|_______| (*8) .
MAPLE
# Maple program: see link.
CROSSREFS
Cf. A174249, A233427, A234312, A234931, A247680, A248102, A249762.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 23 2014
STATUS
approved
A247680
Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L, F.
+10
8
1, 1, 3, 5, 21, 82, 249, 688, 1879, 5690, 17932, 55271, 164427, 485348, 1451110, 4395114, 13313135, 40073992, 120200822, 360897368, 1086543152, 3274191643, 9858847241, 29657925485, 89206237151, 268435863317, 808022052324, 2432169981689, 7319562671432
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 324)
Wikipedia, Pentomino
FORMULA
a(n) ~ c * d^n, where d = 3.009533036298033336764263169394953980849599088993157702490314631810945318907..., c = 0.29272000293879867768013500033525343337565088925220444775140709413075274... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015
EXAMPLE
a(3) = 5:
._____. ._____. ._____. ._____. ._____.
| | | | | |_. | | ._| | | | ._| |_. | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | |_| |_| | | | |_| | | |_| |
|_|_|_| |_|___| |___|_| |_|___| |___|_|.
a(4) = 21:
._______. ._______.
|_. |_. | | ._| ._|
| |_. | | | |_. | |
|_. |_| | | | |_| |
| |___|_| |_| ._| |
|_______| |___|___| ... .
CROSSREFS
Cf. A174249, A233427, A234312, A234931, A247443, A249762, A257866.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 22 2014
STATUS
approved
A247125
Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.
+10
7
1, 0, 2, 1, 16, 10, 59, 60, 330, 397, 1520, 2218, 7875, 12820, 39250, 70045, 202168, 384866, 1038051, 2073580, 5385754, 11156701, 28015232, 59580154, 146333795, 317517636, 766142242, 1686735709, 4019319048, 8946988370, 21116854115, 47386013020, 111065223914
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,1,12,6,2).
FORMULA
G.f.: -1/(2*x^6+6*x^5+12*x^4+x^3+2*x^2-1).
EXAMPLE
a(4) = 16:
._______. ._______. ._______.
| ._____| | ._____| | ._| ._|
|_| |_. | |_| |_. | | | | | |
|_. ._| | |_. ._| | | | | | |
| |_|___| | |_| | | |_| |_| |
|_______| (2) |_____|_| (4) |___|___| (4)
._______. ._______.
| ._____| | ._____|
|_| ._. | |_|_. | |
| |_| |_| | ._| | |
|_____| | | |___| |
|_______| (2) |___|___| (4) .
MAPLE
a:= n-> (<<0|1|0|0|0|0>, <0|0|1|0|0|0>, <0|0|0|1|0|0>,
<0|0|0|0|1|0>, <0|0|0|0|0|1>, <2|6|12|1|2|0>>^n)[6, 6]:
seq(a(n), n=0..40);
CROSSREFS
Cf. A174249, A233427, A234312, A247124, A247268.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 19 2014
STATUS
approved
A264812
Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.
+10
6
1, 1, 3, 5, 13, 52, 123, 366, 909, 2444, 7108, 19157, 53957, 146826, 400704, 1115852, 3059907, 8475420, 23369304, 64225984, 177572352, 488839323, 1349102071, 3722419367, 10255126169, 28303059509, 78013005366, 215160477217, 593488173404, 1636220978049
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
EXAMPLE
a(4) = 13:
._______. ._______. ._______. ._______.
| | | | | | | | | | | | | ._| |
| | | | | | ._| ._| | ._| | | |___| |
| | | | | |_| |_| | |_| | | | | |___|
| | | | | (1) | | | (4) | | | | (6) | ._| | (2)
|_|_|_|_| |___|___| |_ _|_|_| |_|_____| .
a(5) = 52:
._________.
| |_. |
| ._| |___|
|_|_ _| |
| |_| | (2) ...
|_____|___| .
CROSSREFS
Cf. A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A278330.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 25 2015
STATUS
approved
A278330
Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.
+10
6
1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984, 7019, 11148, 35686, 62181, 182776, 339350, 942507, 1841208, 4887096, 9921685, 25442304, 53190380, 132928715, 284198328, 696276202, 1514363221, 3654567764, 8053235650, 19212546163, 42762014028, 101125071372
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,2,8,4,21,-8,-4,-6,0,-16,-8).
FORMULA
G.f.: -(4*x^6+x^3-1) / (8*x^12 +16*x^11 +6*x^9 +4*x^8 +8*x^7 -21*x^6 -4*x^5 -8*x^4 -2*x^3 -2*x^2+1).
a(n) mod 2 = A079978(n).
EXAMPLE
a(2) = 2, a(3) = 1:
.___. .___. ._____.
| | | | | ._. |
| ._| |_. | |_| |_|
|_| | | |_| |_ _|
| | | | | |_| |
|___| |___| |_____| .
MAPLE
a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
[-8, -16, 0, -6, -4, -8, 21, 4, 8, 2, 2, 0][j], 0)))^n.
<<1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984>>)[1, 1]:
seq(a(n), n=0..35);
CROSSREFS
Cf. A079978, A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A264812.
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Nov 18 2016
STATUS
approved
A077909
Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).
+10
5
1, 0, 0, -1, 2, 0, 1, -4, 4, -1, 6, -12, 9, -8, 24, -33, 26, -40, 81, -92, 92, -161, 254, -276, 345, -576, 784, -897, 1266, -1936, 2465, -3060, 4468, -6337, 7990, -10588, 15273, -20664, 26568, -36449, 51210, -67896, 89585, -124108, 170316, -225377, 303278, -418532, 566009, -754032, 1025088
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
._____. ._______. ._______.
| ._. | | ._. | | | | ._. |
|_| |_| |_| |_| | | |_| |_|
|_. ._| , | ._| ._| |_. |_. |
| |_| | | | |_| | | |_| | |
|_____| |_|_____| |_____|_|. - Alois P. Heinz, Jan 03 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,0,-1,2)
FORMULA
a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - Johannes W. Meijer, Aug 28 2013
G.f.: 1/(1 + x^3 - 2*x^4). - Arkadiusz Wesolowski, Nov 20 2013
MAPLE
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
<<1, 0, 0, -1>>)[1, 1]:
seq(a(n), n=0..60); # Alois P. Heinz, Nov 20 2013
MATHEMATICA
CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* Jean-François Alcover, Jun 08 2015, after Arkadiusz Wesolowski *)
PROG
(PARI) Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ Joerg Arndt, Aug 28 2013
CROSSREFS
Partial sums of A077976.
Cf. A174249, A233427, A234312, A234931, A247126.
KEYWORD
sign,easy
AUTHOR
N. J. A. Sloane, Nov 17 2002
STATUS
approved
A257866
Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L.
+10
3
1, 1, 3, 5, 19, 74, 209, 572, 1479, 4304, 13002, 38315, 109651, 308982, 884120, 2560952, 7428183, 21413028, 61433280, 176415916, 507985116, 1464725431, 4220293147, 12145885239, 34945690653, 100586823613, 289649303130, 834087280681, 2401368817168, 6912685066843
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 315)
Wikipedia, Pentomino
FORMULA
a(n) ~ c * d^n, where d = 2.878962978866730659679600165158895088546680936475540731494833253735549346144..., c = 0.33249894796240209167801000207088312509480543003269025485052861968247997... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015
EXAMPLE
a(3) = 5:
._____. ._____. ._____. ._____. ._____.
| | | | | |_. | | ._| | | | ._| |_. | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | |_| |_| | | | |_| | | |_| |
|_|_|_| |_|___| |___|_| |_|___| |___|_|.
a(4) = 19:
._______. ._______.
|_. |_. | | | ._| |
| |_. | | | | | | |
|_. |_| | | | | | |
| |___|_| | |_| | |
|_______| |_|___|_| ... .
CROSSREFS
Cf. A174249, A233427, A234312, A234931, A247443, A247680, A249762.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 11 2015
STATUS
approved
A248102
Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shapes N, Y.
+10
2
1, 0, 0, 18, 24, 238, 842, 4360, 25900, 112178, 613140, 2941170, 14789274, 74895336, 369603312, 1866863986, 9294391952, 46543456838, 233028690018, 1164275409976, 5830080180396, 29149585256266, 145845002931724, 729627382873090, 3649578988919810
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
FORMULA
G.f.: (8*x^15 -52*x^14 -64*x^13 +1087*x^12 -2822*x^11 +2369*x^10 +810*x^9 -2047*x^8 +300*x^7 +122*x^6 +208*x^5 +x^4 +6*x^3 +9*x^2 +2*x -1) / (24*x^15 +4*x^14 -680*x^13 +2673*x^12 -4212*x^11 +2139*x^10 +1574*x^9 -2141*x^8 +456*x^7 -160*x^6 +236*x^5 -11*x^4 +24*x^3 +9*x^2 +2*x -1).
EXAMPLE
a(3) = 18:
._______._._. .___._______. .___._______.
|_. .___| | | |_. |___. ._| |_. |___. ._|
| |_| | ._| | | |_____|_| | | |_____|_| |
| ._| | |_. | | |___. ._| | | |___. |_. |
| | ._|_| |_| | ._| |_|_. | | ._| |___| |
|_|_|_______| (*2) |_|_______|_| (*8) |_|_______|_| (*8) .
CROSSREFS
Cf. A174249, A233427, A234312, A234931, A247680, A247443.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 01 2014
STATUS
approved
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