A366461 a(n) = number of partitions of n that have the maximum number of neighbors; see Comments.

1, 2, 1, 2, 1, 1, 2, 1, 6, 1, 2, 1, 6, 2, 1, 2, 1, 6, 2, 8, 1, 2, 1, 6, 2, 8, 1, 1, 2, 1, 6, 2, 8, 1, 6, 1, 2, 1, 6, 2, 8, 1, 6, 22, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 8, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 8, 30, 1, 2, 1, 6, 2, 8, 1, 6, 22

Offset

1,2

Comments

Partitions p and q of n are neighbors if d(p,q) = 2, where d is the distance function in A366156.

Links

Table of n, a(n) for n=1..86.

Example

Refer to the Example in A366429 to see that a(5) = 1.

Mathematica

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := d[u, v] = Total[Abs[u - v]];
s[n_, k_] := s[n, k] = Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
t[n_] := t[n] = Table[s[n, k], {k, 1, c[n]}]
a[n_] := Max[Map[Length, t[n]]]
b[n_] := b[n] = Select[t[n], Length[#] == a[n] &]
e[n_] := Length[b[n]]
Table[e[n], {n, 1, 24}]

Crossrefs

Cf. A000041, A366156, A366429.

Sequence in context: A309852 A029810 A321601 * A261122 A115660 A128581

Adjacent sequences: A366458 A366459 A366460 * A366462 A366463 A366464

Keyword

nonn

Author

Clark Kimberling, Oct 12 2023

Extensions

More terms from Pontus von Brömssen, Oct 24 2023

Status

approved