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A008483
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Number of partitions of n into parts >= 3.
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43
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1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011.
Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011
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. - E. Keith Lloyd (ekl(AT)soton.ac.uk).
Also the number of regular graphs of degree 2. - Mitch Harris, Jun 22, 2005.
(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 + ...). [Gary W. Adamson, Jun 30 2009]
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, Oct 05 2009]
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REFERENCES
| 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.
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LINKS
| Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 446
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FORMULA
| a(n) = 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.: prod(m>=3, 1/(1-x^m) )
G.f.: sum(n>=0, x^(3*n) / prod(k=1..n, 1-x^k ) ). - Joerg Arndt, Apr 17 2011
a(n) = A121081(n+3) - A121659(n+3). - Reinhard Zumkeller, Aug 14 2006
Euler transformation of A179184. a(n) = A179184(n) + A165652(n). [Jason Kimberley, Jan 05 2011]
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MAPLE
| 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
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MATHEMATICA
| f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 3], {n, 49}] (* Robert G. Wilson v, Jan 31 2011 *)
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PROG
| (MAGMA) p := NumberOfPartitions; A008483 := func< n | n eq 0 select 1 else n le 2 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3)>; - Jason Kimberley, Jan 11 2011.
(PARI) a(n) = numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3) \\ Charles R Greathouse IV, Jul 19 2011
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CROSSREFS
| Essentially the same sequence as A026796.
Contribution from Jason Kimberley, Nov 07 2009 and Jan 05 2011 and Feb 03 2011: (Start)
Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), this sequence (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).
2-regular simple graphs: A179184 (connected), A165652 (disconnected), this sequence (not necessarily connected).
2-regular not necessarily connected graphs without multiple edges [partitions without 2 as a part]: this sequence (no loops allowed [without 1 as a part]), A027336 (loops allowed [parts may be 1]).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), this sequence (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).. (End)
Sequence in context: A091583 A132326 A027195 * A026796 A008925 A036072
Adjacent sequences: A008480 A008481 A008482 * A008484 A008485 A008486
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KEYWORD
| nonn,easy,changed
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AUTHOR
| T. Forbes (anthony.d.forbes(AT)googlemail.com)
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