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A008483 Number of partitions of n into parts >= 3. 54
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; text; internal format)
OFFSET

0,7

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).

For n >= 1, 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. - Jason Kimberley, Oct 05 2009

Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482.  For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], 4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014

REFERENCES

Roland Bacher, P De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2; https://hal.archives-ouvertes.fr/hal-01285685/document

F. Jouneau-Sion, O. Torres, In Fisher's net: exact F-tests in semi-parametric models with exchangeable errors, August 2014, preprint on ResearchGate.

LINKS

Vincenzo Librandi and Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 301 terms from Vincenzo Librandi)

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

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Eric Weisstein's World of Mathematics, Two-Regular Graph

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

a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Feb 26 2015

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, Mar 13 2007

with(combstruct):ZL2:=[S, {S=Set(Cycle(Z, card>2))}, unlabeled]:seq(count(ZL2, size=n), n=0..46); # Zerinvary Lajos, 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, Oct 02 2007

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 *)

Rest[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Length[p]]], {n, 50}]]  (* Clark Kimberling, Feb 27 2014 *)

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

CROSSREFS

Essentially the same sequence as A026796 and A281356.

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: A132326 A027195 * A281356 A026796 A008925 A266749

Adjacent sequences:  A008480 A008481 A008482 * A008484 A008485 A008486

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified July 23 05:58 EDT 2017. Contains 289686 sequences.