A184324 The number of disconnected k-regular simple graphs on 2k+4 vertices.

1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 11, 13, 18, 21, 26, 33, 40, 49, 61, 73, 89, 110, 131, 158, 192, 230, 274, 331, 392, 468, 557, 660, 780, 927, 1088, 1284, 1511, 1775, 2076, 2438, 2843, 3323, 3873, 4510, 5238, 6095, 7057, 8182, 9466, 10945, 12626, 14578, 16780, 19323, 22211

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Formula

a(0)=1. For k>0, a(k) = (k+1) mod 2 + A008483(k+3).

For k>=0, a(k) = A040001(k) + A165652(k+3).

Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(2k+4, k) = C(k+1, k) C(k+3, k) + A000217(C(k+2,k)), from the disconnected Euler transform. C(k+1, k)=1 because K_{k+1} is connected and the unique k-regular graph on k+1 vertices. For k > 1, since D(k+3,k)=0, then C(k+3,k) = E(k+3,k) = E(k+3,2) = A008483(k + 3). Also, for k >0, since D(k+2,k)=0, then C(k+2,k) = E(k+2,k) = E(k+2,1) = (k+1) mod 2. With the examples below and A165652(n)=0 for n < 6 = offset, QED.

Example

The a(0)=1 graph is 4K_1. The a(1)=1 graph is 3K_2. The a(2)=2 graphs are C_3+C_5 and C_4+C_4.

Prog

(Magma) A184324 := func< n | n eq 0 select 1 else (n+1)mod 2 + A008483(n+3) >; // see A008483 for its MAGMA code.

Crossrefs

This sequence is the third highest diagonal of D=A068933: that is a(n)=D(2k+4, k).

Cf. A184325(k) = D(4k+5, 2k) and A184326(k) = D(2k+6, k).

Sequence in context: A027189 A140829 A346523 * A274168 A116575 A244800

Adjacent sequences: A184321 A184322 A184323 * A184325 A184326 A184327

Keyword

nonn,easy

Author

Jason Kimberley, Jan 11 2011

Status

approved