A033691 Minimal number of vertices in 1-1 deficient regular graph where minimal degree is 1 and maximal degree is n.

4, 8, 12, 20, 24, 32, 40, 52, 60, 72, 84, 100, 112, 128, 144, 164, 180, 200, 220, 244, 264, 288, 312, 340, 364, 392, 420, 452, 480, 512, 544, 580, 612, 648, 684, 724, 760, 800, 840, 884, 924, 968, 1012, 1060, 1104, 1152, 1200, 1252, 1300, 1352, 1404, 1460

Offset: 2

Author

Lisa R. Macon (lmacon(AT)valencia.cc.fl.us)

Keywords

• easy
• nice
• nonn

Comments

A 1-1 deficient regular graph is one in which every vertex has the same degree as all of its neighbors except exactly one and its degree differs from the degree of this neighbor by exactly 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Programs

Mathematica

a[n_?EvenQ] := n*(n+2)/2; a[n_ /; Mod[n, 4] == 3] := (n+1)^2/2; a[n_ /; Mod[n, 4] == 1] := (n^2 + 2*n + 5)/2; Table[a[n], {n, 2, 53}] (* Jean-François Alcover, Oct 10 2011 *)

Formula

a(2*i) = 2*i*(i+1) i=1, 2, 3, ...; a(4*i+3) = 2*(i+1)*(4*i+4) i=0, 1, 2, ...; a(4*i+5) = 4*(2*i^2+6*i+5) i=1, 2, 3, ...

Empirical G.f.: -4*x^2*(x^5-2*x^4+x^3+1) / ((x-1)^3*(x+1)*(x^2+1)). [Colin Barker, Jan 15 2013]

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003