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A107953
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Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).
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3
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31, 175, 703, 2415, 7551, 22143, 61951, 167167, 438271, 1122303, 2818047, 6959103, 16941055, 40730623, 96862207, 228130815, 532676607, 1234173951, 2839543807, 6491734015, 14755561471, 33361494015, 75061264383, 168124481535, 375004332031, 833223655423
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OFFSET
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0,1
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COMMENTS
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This sequence is one of a triple sequence A(n,m,l) of the number of fuzzy subsets of a set with n+m+l objects of 3 kinds. There are n,m and l objects for each kind respectively. Here a(n)= A(n,2,1). The sequence A107464 is one other example of A(n,m,l) for m=l=1.
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REFERENCES
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V. Murali, On the number of fuzzy subsets of an (n+3)-element set of specification n^1 2^1 1, Rhodes University Preprint, 2005.
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LINKS
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FORMULA
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a(n) = 2^(n+1) * ((n^2/2)*(n+13) + 21*n + 16) - 1.
G.f.: (48*x^3-120*x^2+104*x-31) / ((x-1)*(2*x-1)^4). - Colin Barker, Jan 15 2015
a(0)=31, a(1)=175, a(2)=703, a(3)=2415, a(4)=7551, a(n)=9*a(n-1)- 32*a(n-2)+ 56*a(n-3)-48*a(n-4)+16*a(n-5). - Harvey P. Dale, Feb 10 2015
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EXAMPLE
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a(3) = 2^4*((9/2)*16 + 21*3 + 16) - 1 = 2415 which is the number of distinct chains in the power set lattice (or fuzzy subsets) of a set X_(n+3) with 3 kinds of objects, n of one kind, 2 of another and one of yet another.
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MATHEMATICA
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Table[2^(n+1) (n^2/2 (n+13)+21n+16)-1, {n, 0, 30}] (* or *) LinearRecurrence[ {9, -32, 56, -48, 16}, {31, 175, 703, 2415, 7551}, 30] (* Harvey P. Dale, Feb 10 2015 *)
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PROG
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(PARI) Vec((48*x^3-120*x^2+104*x-31)/((x-1)*(2*x-1)^4) + O(x^100)) \\ Colin Barker, Jan 15 2015
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CROSSREFS
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KEYWORD
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nice,nonn,easy
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AUTHOR
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Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005
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EXTENSIONS
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a(5) corrected Jun 01 2005
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STATUS
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approved
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