A359971 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.

1, 5, 33, 15, 108, 126, 5, 727, 1031, 38, 2, 1314, 2452, 167, 15, 2, 6811, 12102, 988, 52, 14904, 27626, 3255, 214, 4, 2, 2, 39172, 73289, 10062, 795, 19, 1, 65833, 127951, 18476, 1464, 64, 5, 201643, 370880, 59630, 5548, 250, 7, 2, 288196, 541258, 91037, 9692, 428, 20, 4, 741597, 1351301, 239180, 27510, 1434, 58

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

Keywords

• nonn
• tabf

Comments

The number of vertices along the shorter edges is A005728(n).

No formula for a(n) is known. The sequence is inspired by the Farey fan; see A360042.

See A359968 and A359969 for images of the triangle.

Examples

			The table begins:
        1;
        5;
       33,      15;
      108,     126,      5;
      727,    1031,     38,     2;
     1314,    2452,    167,    15,    2;
     6811,   12102,    988,    52;
    14904,   27626,   3255,   214,    4,   2, 2;
    39172,   73289,  10062,   795,   19,   1;
    65833,  127951,  18476,  1464,   64,   5;
   201643,  370880,  59630,  5548,  250,   7, 2;
   288196,  541258,  91037,  9692,  428,  20, 4;
   741597, 1351301, 239180, 27510, 1434,  58;
  1095197, 2025237, 374907, 44880, 2491, 104, 4, 2;
  1747260, 3279178, 628335, 76787, 4600, 178, 6;
  ...
		

Crossrefs

Cf. A359968 (vertices), A359969 (regions and row sums), A359970 (edges), A005728, A360042, A359977, A359694, A358951, A358889.