The On-Line Encyclopedia of Integer Sequences (OEIS)

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Revisions by Travis Hoppe (See also Travis Hoppe's wiki page)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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Number of 4-fold rotationally symmetric pentomino tilings of the 5n X 5n square such that no two tiles can be replaced with two pentomominoes in a new configuration.
(history; published version)

#7 by Travis Hoppe at Fri Sep 29 12:32:44 EDT 2023

STATUS

editing

proposed

#6 by Travis Hoppe at Thu Sep 21 15:12:13 EDT 2023

LINKS

Travis Hoppe, <a href="/A365862/a365862.py.txt">Python script to compute the sequence</a>,

Python Travis Hoppe, <a href="/OEIS_A365862.py">script</a> to compute the sequence. <a href="/ray_intersect_1_over_0005a365862.png">Visual depiction</a> of the positive x-y quadrant for a(5)</a>

Travis Hoppe, <a href="/A365862/a365862.py.txt">TITLE FOR LINK</a>

Travis Hoppe, <a href="/A365862/a365862.png">TITLE FOR LINK</a>

Discussion

Thu Sep 21

18:50

Andrew Howroyd: 8 * A124254.

18:55

Andrew Howroyd: (How does a radius of 1 make sense?  Wouldn't the circles overlap? For n>=2, terms are obviously divisible by 8)

23:26

Travis Hoppe: @Andrew, oh it completely is a copy of A124254! I searched for it divided by 4 by not by 8 (the full symmetry). Even the example I came up with is similar. How can I delete this entry?

Thu Sep 28

23:40

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A365862 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

Fri Sep 29

12:32

Travis Hoppe: EDITORS: this sequence is a duplicate of A124254 (up to a factor of 8). While I calculated the terms by considering the geometry and A124254 makes a conjecture, I believe that these are the same. I can not remove this draft as a submitter, but I believe the editors can. Thank you for considering it!

#5 by Travis Hoppe at Thu Sep 21 11:00:41 EDT 2023

DATA

4, 8, 16, 32, 40, 72, 88, 120, 152, 192, 224, 264, 304, 384, 440, 480, 536, 616, 672, 768, 832, 928, 1000, 1112, 1184, 1280, 1384, 1488

LINKS

Python <a href="/OEIS_A365862.py">script</a> to compute the sequence. <a href="/ray_intersect_1_over_0005.png">Visual depiction</a> of the positive x-y quadrant for a(5)

Travis Hoppe, <a href="/A365862/a365862.py.txt">TITLE FOR LINK</a>

Travis Hoppe, <a href="/A365862/a365862.png">TITLE FOR LINK</a>

EXAMPLE

For n=3 we the a(3)=16 circles are computed by counting 3 the circles three in the positive x-y quadrant at (1,1),(1,2),(2,1). There By symmetry there are 4 quadrants and then 4 circles on the axis at (0,1),(1,0),(-1,0),(0,-1) giving 4*3+4 solutions.

For n=4 the a(4)=32 circles are similar to a(3) but there are additional circles in positive x-y quadrant circles are additionally at (1,3),(1,4),(3,1),(4,1),(2,3),(3,2) but not . There is no circle at (2,2) as it is occluded by the circle at (1,1).

EXTENSIONS

Computed more terms. Responded to comments by editors. Added code to compute the sequence and an image of a(5).

Discussion

Thu Sep 21

11:04

Travis Hoppe: @Stefano added the code and an image. Not sure if I got the links correct. 
@Michel adjusted the explanation text as requested.
@Joerg this is a different sequence from A175341. This sequence counts the number of grid points within a circle. This new sequence counts looks at the number of grid points that can be seen from the origin for a fixed radius. For example at a(5), the circle at (2,2) is not counted in the sequence because it is occluded by the circle at (1,1).

#3 by Travis Hoppe at Thu Sep 21 00:18:13 EDT 2023

STATUS

editing

proposed

Discussion

Thu Sep 21

00:21

Stefano Spezia: Keyword more is necessary. Example should end with a “.”. Your code could be added in Link as a text file

#2 by Travis Hoppe at Thu Sep 21 00:16:14 EDT 2023

NAME

allocated for Travis HoppeNumber of circles of radius 1/n on 2D grid points visible from the origin

DATA

4, 8, 16, 32, 40, 72, 88, 120, 152, 192, 224, 264, 304, 384, 440, 480, 536, 616, 672, 768, 832, 928, 1000, 1112

OFFSET

1,1

COMMENTS

A circle is visible if one can draw a ray from the origin to any point on the circle without first intersecting any other circles. There is no circle at the origin.

EXAMPLE

For n=3 we a(3)=16 circles by counting 3 in the positive x-y quadrant at (1,1),(1,2),(2,1). There are 4 quadrants and then circles at (0,1),(1,0),(-1,0),(0,-1) giving 4*3+4 solutions. For n=4 the positive x-y quadrant circles are additionally at (1,3),(1,4),(3,1),(4,1),(2,3),(3,2) but not at (2,2) as it is occluded by the circle at (1,1)

KEYWORD

allocated

nonn

AUTHOR

Travis Hoppe, Sep 20 2023

STATUS

approved

editing

Discussion

Thu Sep 21

00:17

Travis Hoppe: I'm not sure if there is better language for "visible" here, welcome to edits. I've solved the first 24 terms using sympy, but the python code is a bit long to post.

#1 by Travis Hoppe at Thu Sep 21 00:16:14 EDT 2023

NAME

allocated for Travis Hoppe

KEYWORD

allocated

STATUS

approved

a(n) is the number of distinct lengths between consecutive points of the Farey sequence of order n.
(history; published version)

#20 by Travis Hoppe at Fri Jul 15 17:00:00 EDT 2022

EXAMPLE

For n=5, the Farey sequence (completely reduced fractions between 0 and 1, with denominators less than or equal to n) is [0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1]. The distinct lengths between consecutive points are {1/5, 1/20, 1/12, 1/15, 1/10} so a(5) = 5.

Discussion

Fri Jul 15

17:00

Travis Hoppe: @Kevin sounds good. Adjusted. Feel free to make edits to help with readability!

#19 by Travis Hoppe at Thu Jul 14 23:42:38 EDT 2022

NAME

a(n) is the number of distinct lengths between consecutive points when the interval [0, 1] is partitioned by of the rationals a/b where 0 < a < b < Farey sequence of order n.

EXAMPLE

For n=5, the distinct points on the unit interval are in order Farey sequence (completely reduced fractions between 0 and 1, with denominators less than or equal to n) is [0, /1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1]. The distinct lengths between consecutive points are {1/5, 1/20, 1/12, 1/15, 1/10} so a(5) = 5.

STATUS

proposed

editing

Discussion

Fri Jul 15

04:34

Kevin Ryde: "completely reduced ..." is good for a comment.  (Examples not really the place for explanation of definitions.)

#18 by Travis Hoppe at Thu Jul 14 09:35:11 EDT 2022

STATUS

editing

proposed

Discussion

Thu Jul 14

19:24

Kevin Ryde: NAME says b < n but example includes b=n ?

19:25

Kevin Ryde: If code compute(n) is a(n) then it can be helpfully named a(n) for consistency with the text.

19:29

Kevin Ryde: Do you particularly need "partitioned" in the name?  Just between consecutive rationals a/b etc etc?

19:31

Kevin Ryde: About 0 and 1, I'd be tempted to include them in the other a/b.  Whatever that would be, 1 <= b <= n and 0 <= a <= b ... if that's right.

23:34

Travis Hoppe: @Kevin looking at the series A005728, these terms we are trying to describe are just the Farey series of order n. I think it's a clearer to just use that.

#17 by Travis Hoppe at Thu Jul 14 09:33:13 EDT 2022

NAME

a(n) is the number of distinct lengths between consecutive points when the unit interval [0, 1] is partitioned by the rationals a/b where 0 < a < b < n.

Discussion

Thu Jul 14

09:33

Travis Hoppe: @Kevin adjust the name to be more explicit that the endpoints [0, 1] are included.

09:34

Travis Hoppe: OEIS editors: I believe I've addressed and updated the entry for all points made by the reviewers (thank you all!). Submitting the sequence now.