cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A187781 Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 14, 14, 25, 21, 41, 40, 63, 60, 92, 72, 129, 121, 175, 166, 231, 192, 298, 285, 377, 360, 469, 350, 575, 553, 696, 666, 833, 744, 987, 956, 1159, 1123, 1350, 1165, 1561, 1508, 1793, 1741, 2047, 1875, 2324, 2255, 2625, 2563, 2951, 2761, 3303, 3214, 3682, 3588, 4089, 3695
Offset: 3

Views

Author

Martin Renner, Jan 05 2013

Keywords

Examples

			a(5) = 3 since the 11 regions of a regular pentagon with all diagonals drawn consist of three different noncongruent polygons: two different triangles (each 5 times) and 1 pentagon.
a(6) = 3 since the 24 regions of the regular hexagon with all diagonals drawn consist of three different noncongruent polygons: 2 triangles (one 6 times, one 12 times) and 1 quadrilateral (6 times).
a(7) = 7 since the 50 regions of the regular heptagon with all diagonals drawn consist of seven different noncongruent polygons: 4 triangles (three 7 times, one 14 times), 1 quadrilateral (7 times), 1 pentagon (7 times) and 1 heptagon.

Links

Crossrefs

Cf. A006561, A007678, A165217, A187782.

Extensions

Corrected a(12) and a(16), extended from a(18) through a(60), corrected small typo in a(7) example - Christopher Scussel, Jun 23 2023

A320422 The number of tiles inside a regular n-gon created by diagonals that run from each of the n vertices to the n-2 midpoints of opposite edges.

Original entry on oeis.org

6, 25, 50, 145, 224, 497, 684, 1281, 1650, 2713, 3406, 5223, 6300, 9137, 10744, 14779, 17214, 23161, 26250, 34277, 38456, 49105, 54500, 68225, 75114, 92457, 101094, 122371
Offset: 3

Views

Author

R. J. Mathar, Jan 08 2019

Keywords

Comments

Sequence proposed by Thomas Young: draw the regular n-gon and construct n*(n-2) diagonals that run from each of the n vertices to the n-2 points in the middle of the opposite edges, where opposite edges are those not adjacent to the vertex. Count the non-overlapping polygons inside the n-gon which have sides that are sections of the diagonals or sections of the n-gon edges.

Links

Crossrefs

Cf. A165217, A320431.

Formula

Conjecture: a(2n+1) = n*(2*n+1)*(4*n^2-3*n+5)/3. - Thomas Young (tyoung(AT)district16.org), Jan 05 2019

A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

Original entry on oeis.org

1, 1, 31, 13, 71, 25, 127, 41, 199, 61, 287, 85, 391, 113, 511, 145, 647, 181, 799, 221, 967, 265, 1151, 313, 1351, 365, 1567, 421, 1799, 481, 2047, 545, 2311, 613, 2591, 685, 2887, 761, 3199, 841, 3527, 925, 3871, 1013, 4231, 1105, 4607, 1201, 4999, 1301, 5407, 1405, 5831, 1513, 6271, 1625, 6727, 1741
Offset: 3

Views

Author

R. J. Mathar, Jan 08 2019

Keywords

Comments

Sequence proposed by Thomas Young: draw the regular n-gon and construct 2*n lines that run from both ends of the n edges perpendicular into the n-gon until they hit an opposite edge. (For n even the lines actually hit another vertex, so there are only n additional lines). a(n) is the number of non-overlapping tiles inside the n-gon with edges that are sections of the lines or n-gon edges.

Links

Crossrefs

Cf. A165217, A320422

Formula

a(2n) = 2*n^2+2*n+1 = A001844(n), n>1. a(2n+1) = 8*n^2-1 = A157914(n), n>1. - Thomas Young (tyoung(AT)district16.org), Nov 11 2017
G.f.: x^3 +x^4 -x^5*(31+13*x-22*x^2-14*x^3+7*x^4+5*x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Jan 21 2019
a(n) = 1+n*A064680(n-2), n>=5. - R. J. Mathar, Jan 21 2019
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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