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A026799
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Number of partitions of n in which the least part is 6.
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14
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0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,19
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COMMENTS
| a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.
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LINKS
| Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
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FORMULA
| G.f.: x^6 * Product 1/(1-x^m); m=6..inf.
a(n) = p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)-p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) for n>0 where p(n)=A000041(n). [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Oct 28 2010]
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EXAMPLE
| a(0)=0 because there does not exist a least part of the empty partition.
The a(6)=1 partition is 6.
The a(12)=1 partition is 6+6.
The a(13)=1 partition is 6+7.
.............................
The a(17)=1 partition is 6+11.
The a(18)=2 partitions are 6+6+6 and 6+12.
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MAPLE
| ZL := [ B, {B=Set(Set(Z, card>=6))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..63); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007
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MATHEMATICA
| f[1, 1] = f[0, k_] = 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, 6], {n, 0, 50}] (* Rorbert G. Wilson v, Jan 31 2011 *)
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PROG
| (MAGMA) p := func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // Jason Kimberley, Feb 04 2011
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CROSSREFS
| 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), A008483 (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 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 04 2011
Sequence in context: A026824 A025149 * A185326 A027190 A036824 A108104
Adjacent sequences: A026796 A026797 A026798 * A026800 A026801 A026802
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001.
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