0,7

By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3.

For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message.

Also the number of regular graphs of degree 2. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) Jun 22, 2005.

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 30 2009: (Start)

(1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2x^6 + ...) = (1 + x + 2x^2 + 3x^3 + 5x^4 + ...)

* 1 / (1 + x + 2x^2 + 2x^3 + 3x^4 + 3x^5 + 4x^6 + 4x^7 + ...). (End)

Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. [From Jason Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Oct 05 2009]

R.-Q. Feng, J. H. Kwak and E. K. Lloyd, Isomorphism classes of authentication codes, Bull. Austral. Math. Soc. 69 (2004), no. 2, 203-215.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 446

p(n) - p(n - 1) - p(n - 2) + p(n - 3) where p(n) is the number of unrestricted partitions of n into positive parts (A000041).

G.f.: Product 1/(1-x^m); m=3..inf.

a(n) = A121081(n+3) - A121659(n+3). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 14 2006

series(1/product((1-x^i), i=3..50), x, 51);

ZL := [ B, {B=Set(Set(Z, card>=3))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

with(combstruct):ZL2:=[S, {S=Set(Cycle(Z, card>2))}, unlabeled]:seq(count(ZL2, size=n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2007

with (combstruct):a:=proc(m) [A, {A=Set(Cycle(Z, card>m))}, unlabeled]; end: A008483:=a(2):seq(count(A008483, size=n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007

Essentially the same sequence as A026796.

Regular graphs A005176 (any degree), A051031 (triangular array), specified degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).

Sequence in context: A091583 A132326 A027195 this_sequence A026796 A008925 A036072

Adjacent sequences: A008480 A008481 A008482 this_sequence A008484 A008485 A008486

nonn,new

T. Forbes (anthony.d.forbes(AT)googlemail.com)

Additional comments from E. Keith Lloyd (ekl(AT)soton.ac.uk).

Regular graphs cross-references edited by Jason Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Nov 07 2009