A393654 Triangle read by rows: T(n,k), for n >= 4 and 3 <= k < n, is the number of irreducible meanders with n intersections and k tangential intersections.

0, 2, 2, 0, 0, 0, 0, 0, 12, 8, 0, 10, 16, 14, 8, 4, 12, 36, 48, 72, 36, 0, 0, 0, 210, 240, 162, 72, 2, 8, 180, 360, 884, 884, 530, 212, 0, 50, 160, 798, 1368, 3744, 3328, 1452, 528, 4, 20, 300, 800, 5964, 8946, 14950, 11960, 4314, 1438, 0, 40, 160, 3626, 8288, 31320, 41760, 57904, 42112, 12402, 3816

Offset

4,2

Comments

An irreducible meander is a type of singular meander that has no nontrivial submeanders.

Links

Yury Belousov, Table of n, a(n) for n = 4..139 (rows n = 4..19, flattened)

Yury Belousov, Singular meanders, Zap. Nauchn. Sem. POMI, 549 (2025), 49-64.

Yury Belousov, Prime Factorization of Meanders, arXiv:2112.10289 [math.CO], 2025.

Yury Belousov, C++ code for generating the sequence, GitHub.

Formula

T(n, k) = 0 for k < 3 and for k = n;

T(2*m, 3) = 0 for m >= 2;

T(2*m+1, 3) = phi(m+3) - 2 for m >= 1, where phi(x) is Euler's totient function;

T(2*m+1, 4) = (m-1)*(phi(m+3) - 2) for m >= 2;

Sum_{k >= 0} T(k+1, k)*t^k = (1/8)*(-1 + ((1 - 3*t)/(1 + t))^(1/2))^4;

Sum_{k >= 0} T(k+2, k)*t^k = -(2*t - 1 + ((1-3*t)/(1+t))^(1/2))/((1-3*t)*(1+t)^5)^(1/2).

Example

The triangle T(n,k), n >= 4, 3 <= k < n, begins:
  0;
  2, 2;
  0, 0, 0;
  0, 0, 12, 8;
  0, 10, 16, 14, 8;
  4, 12, 36, 48, 72, 36;
  ...

Crossrefs

Sequence in context: A281084 A186230 A214304 * A248640 A376562 A281083

Adjacent sequences: A393651 A393652 A393653 * A393655 A393656 A393657

Keyword

nonn,tabl,hard

Author

Yury Belousov, Mar 24 2026

Status

approved