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.

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A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 1, 35, 40, 8, 1, 126, 140, 20, 0, 1, 330, 228, 60, 12, 0, 1, 715, 644, 112, 0, 0, 0, 1, 1365, 1168, 208, 0, 0, 0, 0, 1, 2380, 1512, 216, 54, 54, 0, 0, 0, 1, 3876, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 5985, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1, 8855, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 12, 12650
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, May 23 2020

Keywords

Examples

			Table begins:
      0;
      0;
      0;
      1;
      5;
     12,    1;
     35;
     40,    8,   1;
    126;
    140,   20,   0,   1;
    330;
    228,   60,  12,   0,   1;
    715;
    644,  112,   0,   0,   0,  1;
   1365;
   1168,  208,   0,   0,   0,  0, 1;
   2380;
   1512,  216,  54,  54,   0,  0, 0, 1;
   3876;
   3360,  480,   0,   0,   0,  0, 0, 0, 1;
   5985;
   5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
   8855;
   6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
  12650;
  11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
  17550;
  15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
  23751;
  13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
  31465;
  28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  40920;
  37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  52360;

Links

Crossrefs

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A007569, A007678, A053126, A292105, A333275.

Formula

If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).

A371264 Irregular triangle read by rows: T(n,k) is the number of internal vertices in the graph A371254(n) that are created by the crossing of k arcs, with k>=2.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 5, 0, 0, 0, 0, 1, 49, 14, 48, 8, 171, 27, 0, 0, 0, 0, 0, 1, 190, 20, 484, 55, 360, 12, 0, 0, 12, 0, 0, 0, 0, 0, 1, 1027, 91, 1078, 70, 1830, 120, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2000, 112, 3052, 204, 3114, 90, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Scott R. Shannon, Mar 18 2024

Keywords

Comments

See A371254 for images of the graphs.

Examples

			The table begins:
0;
0;
0, 1;
0;
5, 5;
0, 0, 0, 0, 1;
49, 14;
48, 8;
171, 27, 0, 0, 0, 0, 0, 1;
190, 20;
484, 55;
360, 12, 0, 0, 12, 0, 0, 0, 0, 0, 1;
1027, 91;
1078, 70;
1830, 120, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0 1;
2000, 112;
3052, 204;
3114, 90, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
5662, 285;
5740, 240;
8610, 378, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
8888, 330;
12995, 506;
12312, 336, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18650, 650;
18668, 572;
25596, 810, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                        \\ 0,  1;
25928, 728;
34887, 1015;
32580, 510, 0, 0, 150, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                             \\ 0, 0, 0, 0, 0, 1;
46097, 1240;
46464, 1120;
.
.

Crossrefs

Cf. A371254, A292105, A335102, A292104.

Formula

Sum of row(n) = A371254(n) - n;

A383462 Triangle read by rows: T(n,k) (n >= 3, 2 <= k <= n-1) = number of vertices where k lines cross in the planar graph formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

3, 1, 4, 10, 0, 5, 30, 1, 0, 6, 84, 0, 0, 0, 7, 120, 16, 1, 0, 0, 8, 324, 0, 0, 0, 0, 0, 9, 420, 40, 0, 1, 0, 0, 0, 10, 880, 0, 0, 0, 0, 0, 0, 0, 11, 708, 156, 24, 0, 1, 0, 0, 0, 0, 12, 1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

Keywords

Comments

For other illustrations see A146212, A344857, A292105.

Examples

			Triangle begins:
   3;
   1, 4;
   10, 0, 5;
   30, 1, 0, 6;
   84, 0, 0, 0, 7;
   120, 16, 1, 0, 0, 8;
   324, 0, 0, 0, 0, 0, 9;
   420, 40, 0, 1, 0, 0, 0, 10;
   880, 0, 0, 0, 0, 0, 0, 0, 11;
   708, 156, 24, 0, 1, 0, 0, 0, 0, 12;
   1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
   1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14;
   3780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15;
   3408, 544, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 16;
   6664, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17;
   4572, 756, 108, 108, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 18;
   10944, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19;
   9840, 1280, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 20;
   .
   .
See the attached table for rows 3 to 100.
For n = 8, we may classify the vertices by degree and according to whether they are outside, on, or inside the octagon:
                V2      V3      V4      V5      V6      V7
----------------------------------------------------------
   outside      80      8
   on           0       0       0       0       0       8
   inside       40      8       1       0       0       0
----------------------------------------------------------
   totals       120     16      1       0       0       8
----------------------------------------------------------
   Grand total: 145 = A146212(8)
In general, for n >= 3, the counts for inside the defining polygon are given by row n of A292105, the total number on or inside the polygon by A007569, and the number outside by A146213.

Links

Crossrefs

Row sums are A146212.
Cf. A007569, A146213, A292105, A344857.
Previous Showing 11-13 of 13 results.

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

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