A393352 Number of circular parking functions of length n avoiding the patterns 123 and 132.

1, 1, 2, 4, 9, 16, 37, 64, 137, 262, 531, 1024, 2101, 4096, 8261, 16426, 32913, 65536, 131401, 262144, 524861, 1048720, 2098185, 4194304, 8391001, 16777276, 33558539, 67109392, 134226093, 268435456, 536888491, 1073741824, 2147516705, 4294969372, 8590000143, 17179869526

Offset

0,3

Links

Michael De Vlieger, Table of n, a(n) for n = 0..3322

Lara Pudwell, Pattern avoidance in cyclic parking functions, 2025.

Lara Pudwell, Pattern Avoidance in Circular Parking Functions, Valparaiso Univ. (2026). See pp. 11 (Table 3), 13 (Theorem 16).

Formula

a(n) = (Sum_{i=1..n} 2^gcd(i, n))/2 - (n-1).

Example

For n=4, the a(4)=9 parking functions are 1111, 1112, 1113, 1114, 1122, 1133, 1212, 1222, 1313.

Maple

a := n -> 1/2*add(2^igcd(i, n), i = 1 .. n) + 1 - n:
seq(a(n), n = 1 .. 35);

Mathematica

Table[Sum[2^GCD[i, n], {i, n}]/2 - (n - 1), {n, 0, 35}] (* Michael De Vlieger, Mar 13 2026 *)

Crossrefs

Cf. A019536, A121774, A171005, A213937, A394236, A394237, A394238.

Sequence in context: A056336 A056331 A203318 * A282776 A283015 A264629

Adjacent sequences: A393349 A393350 A393351 * A393353 A393354 A393355

Keyword

nonn

Author

Lara Pudwell, Mar 13 2026

Status

approved