A368002 -id:A368002

Search: a368002 -id:a368002

Displaying 1-4 of 4 results found.

page 1

A368000

a(n) is the numerator of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears as the image of a simple random walk on the square lattice.

+10
8

1, 1, 1, 1, 1, 4, 1, 1, 1, 97, 1, 1, 1, 8, 1, 1, 8, 8, 1, 1, 1, 867, 9565, 1, 1, 2495, 1, 262781, 389, 9565, 389, 262781, 1, 867, 1, 597, 389, 1, 631381, 597, 389, 1, 1, 389, 1, 597, 1, 1, 389, 597, 389, 1, 597, 2501, 412, 1, 2635, 1706571966622, 1706571966622, 1117, 1117

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

OFFSET

1,6

COMMENTS

In a simple random walk on the square lattice, draw a unit square around each visited point. a(n)/A368001(n) is the probability that, when the appropriate number of distinct points have been visited, the drawn squares form a particular one of the fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1).

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

LINKS

Table of n, a(n) for n=1..61.

Index entries for sequences related to polyominoes.

FORMULA

a(n)/A368001(n) = (A367994(n)/A367995(n))/A335573(n+1).

EXAMPLE

As an irregular triangle:

1, 1;

1, 4, 1, 1, 1;

97, 1, 1, 1, 8, 1, 1, 8, 8, 1, 1, 1;

...

CROSSREFS

Cf. A000105, A246521, A335573, A367675, A367764, A367994, A367995, A368001 (denominators), A368002, A368004.

KEYWORD

nonn,frac,tabf

AUTHOR

Pontus von Brömssen, Dec 09 2023

STATUS

approved

A367996

Numerator of the least probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

+10
6

1, 1, 1, 2, 1, 1648, 10916, 227056, 17, 37138, 32596907, 203911047902268383, 61

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

OFFSET

1,4

COMMENTS

a(n) is the numerator of the minimum of A367994/A367995 over the n-th row. See A367994 for details.

LINKS

Table of n, a(n) for n=1..13.

Pontus von Brömssen, Illustration of all least probable polyominoes with at most 13 cells.

Index entries for sequences related to polyominoes.

CROSSREFS

Cf. A367994, A367995, A367997 (denominators), A367998, A368002.

KEYWORD

nonn,frac,more

AUTHOR

Pontus von Brömssen, Dec 08 2023

STATUS

approved

A368004

Numerator of the greatest probability that a particular fixed polyomino with n cells appears as the image of a simple random walk on the square lattice.

+10
6

1, 1, 1, 4, 97, 2495, 98576101, 790070277194753299070819, 1697817285476742288131092, 301424494727669492958807965129775458632594691220000993251280413656197020195992465248816242330162

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

OFFSET

1,4

COMMENTS

a(n) is the numerator of the maximum of A368000/A368001 over the n-th row. See A368000 for details.

LINKS

Pontus von Brömssen, Table of n, a(n) for n = 1..13

Index entries for sequences related to polyominoes.

EXAMPLE

For 1 <= n <= 13, the following are all polyominoes, up to reflections and rotations, that have the maximum probabilities for their respective sizes. Except for n = 3, there is just one such polyomino (again, up to reflections and rotations).

_ _

_ _ | | _ _ | |_

_ | | | |_ | | | | | |

|_| |_| |_ _| |_| |_ _| |_ _|

_ _ _ _ _ _ _ _ _

| | _| | | |_ | |

| | | _| | | | |

|_ _| |_ _| |_ _ _| |_ _ _|

_ _ _ _ _ _ _ _ _

_| | | |_ | | _| |_

| | | | | | | |

| _| | | | | | _|

|_ _| |_ _ _| |_ _ _| |_ _ _|

CROSSREFS

Cf. A367677, A367766, A367998, A368000, A368001, A368002, A368005 (denominators).

KEYWORD

nonn,frac

AUTHOR

Pontus von Brömssen, Dec 21 2023

STATUS

approved

A368003

Denominator of the least probability that a particular fixed polyomino with n cells appears as the image of a simple random walk on the square lattice.

+10
4

1, 2, 6, 28, 1001, 291291, 243373, 1634249695, 32872944939783505504120, 29741903410, 800946519853683297326312, 124513191408778015284779022326574057102628201495649607240, 4065704197026943714470537818579676

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

OFFSET

1,2

COMMENTS

a(n) is the denominator of the minimum of A368000/A368001 over the n-th row. See A368000 for details.

LINKS

Table of n, a(n) for n=1..13.

Pontus von Brömssen, Illustration of all least probable polyominoes with at most 13 cells.

Index entries for sequences related to polyominoes.

CROSSREFS

Cf. A367997, A368000, A368001, A368002 (numerators), A368005.

KEYWORD

nonn,frac,more

AUTHOR

Pontus von Brömssen, Dec 09 2023

STATUS

approved

page 1

Search completed in 0.004 seconds