A184324 - OEIS

%I #14 Sep 08 2022 08:45:55

%S 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,

%T 230,274,331,392,468,557,660,780,927,1088,1284,1511,1775,2076,2438,

%U 2843,3323,3873,4510,5238,6095,7057,8182,9466,10945,12626,14578,16780,19323,22211

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

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

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

%F 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.

%e 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.

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

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

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

%K nonn,easy

%O 0,3

%A _Jason Kimberley_, Jan 11 2011