A333136 -id:A333136

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A332978

The number of regions formed inside a triangle with leg lengths equal to the Pythagorean triples by straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

+10
4

271, 5746, 14040, 32294, 50551, 108737, 180662, 276533, 259805, 558256, 591687, 901811, 1117126, 1015277, 1386667, 1223260, 1944396, 3149291, 3165147, 4523784, 4764416, 4859839, 6025266, 7186096

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

OFFSET

1,1

COMMENTS

The terms are from numeric computation - no formula for a(n) is currently known.

LINKS

Table of n, a(n) for n=1..24.

Scott R. Shannon, Triangle regions for leg lengths (3,4,5).

Scott R. Shannon, Triangle regions for leg lengths (6,8,10).

Scott R. Shannon, Triangle regions for leg lengths (5,12,13).

Scott R. Shannon, Triangle regions for leg lengths (9,12,15).

Scott R. Shannon, Triangle regions for leg lengths (8,15,17).

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Wikipedia, Pythagorean triple.

EXAMPLE

The triples are ordered by the total sum of the leg lengths:

Triple | Number of regions

(3, 4, 5) | 271

(6, 8, 10) | 5746

(5, 12, 13) | 14040

(9, 12, 15) | 32294

(8, 15, 17) | 50551

(12, 16, 20) | 108737

(7, 24, 25) | 180662

(15, 20, 25) | 276533

(10, 24, 26) | 259805

(20, 21, 29) | 558256

(18, 24, 30) | 591687

(16, 30, 34) | 901811

(21, 28, 35) | 1117126

(12, 35, 37) | 1015277

(15, 36, 39) | 1386667

(9, 40, 41) | 1223260

(24, 32, 40) | 1944396

(27, 36, 45) | 3149291

(14, 48, 50) | 3165147

(20, 48, 52) | 4523784

(24, 45, 51) | 4764416

(30, 40, 50) | 4859839

(28, 45, 53) | 6025266

(33, 44, 55) | 7186096

CROSSREFS

Cf. A333135 (n-gons), A333136 (vertices), A333137 (edges), A103605 (Pythagorean triple ordering), A007678, A092867, A331452.

KEYWORD

nonn,more

AUTHOR

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

EXTENSIONS

a(8)-a(24) from Lars Blomberg, Jun 07 2020

STATUS

approved

A333135

Irregular table read by rows: Take a triangle with Pythagorean triple leg lengths with all diagonals drawn, as in A332978. Then T(n,k) = number of k-sided polygons in that figure for k >= 3 where the legs are divided into unit length parts.

+10
4

139, 94, 34, 3, 1, 2383, 2421, 760, 167, 13, 2, 5307, 5958, 2113, 563, 80, 17, 2, 13083, 13560, 4479, 1002, 153, 16, 1, 18827, 20896, 8256, 2139, 377, 49, 6, 1, 42992, 45400, 15930, 3771, 579, 60, 5, 63526, 79275, 28922, 7315, 1404, 202, 14, 4

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

OFFSET

1,1

COMMENTS

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..213 (the first 24 rows)

EXAMPLE

Table begins:

139, 94, 34, 3, 1;

2383, 2421, 760, 167, 13, 2;

5307, 5958, 2113, 563, 80, 17, 2;

13083, 13560, 4479, 1002, 153, 16, 1;

18827, 20896, 8256, 2139, 377, 49, 6, 1;

42992, 45400, 15930, 3771, 579, 60, 5;

63526, 79275, 28922, 7315, 1404, 202, 14, 4;

The row sums are A332978.

CROSSREFS

Cf. A332978 (regions), A333136 (vertices), A333137 (edges), A103605 (Pythagorean triple ordering), A007678, A092867, A331452.

KEYWORD

nonn,tabf

AUTHOR

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

EXTENSIONS

Corrected typo in a(12) and a(49) and beyond from Lars Blomberg, Jun 07 2020

STATUS

approved

A333137

The number of edges formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

+10
3

500, 10883, 27220, 61570, 98657, 208739, 353922, 533442, 507744, 1100007, 1146403, 1771007, 2168628, 2002321, 2719907, 2413390, 3787444, 6140737, 6238486, 8906032, 9394871, 9495582, 11939407, 14063303

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

OFFSET

1,1

COMMENTS

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

LINKS

Table of n, a(n) for n=1..24.

CROSSREFS

Cf. A332978 (regions), A333135 (n-gons), A333136 (vertices), A103605 (Pythagorean triple ordering), A274586 , A332600, A331765.

KEYWORD

nonn,more

AUTHOR

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

EXTENSIONS

a(8)-a(24) from Lars Blomberg, Jun 07 2020

STATUS

approved

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