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

1,2

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.

Links

Table of n, a(n) for n=1..59.

Example

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.

Sequence in context: A349099 A174471 A275233 * A043068 A194405 A299518

Adjacent sequences: A359968 A359969 A359970 * A359972 A359973 A359974

Keyword

nonn,tabf

Author

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

Status

approved