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A190531
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Number of idempotents in Identity Difference Partial Transformation semigroup.
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0
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2, 5, 17, 57, 185, 593, 1901, 6121, 19793, 64161, 208085, 674105, 2179001, 7023409, 22566269, 72268809, 230696609, 734153537, 2329503653, 7371475033, 23267249417, 73268609745, 230224239437, 721965697577, 2259855722225
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OFFSET
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1,1
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COMMENTS
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IDP_n is a semigroup with the non-isolation property and E(IDP_n) denotes the set of idempotents (satisfying e^2 = e) in IDP_n.
#E(IDP_n) is the number of idempotent elements in the semigroup IDP_n for each n in N. E(IDP_n) is a subset of partial transformation semigroup having the property that the difference in the image, Im(alpha), is not greater than 1 and e^2 = e for each e in IDP_n.
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LINKS
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FORMULA
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#IDP_n = (n-1)*3^(n-2) + n*2^(n-1) - n + 2.
G.f.: -x*(-2+19*x-73*x^2+145*x^3-153*x^4+68*x^5) / ( (x-1)^2*(3*x-1)^2*(2*x-1)^2 ). - R. J. Mathar, Jun 19 2011
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EXAMPLE
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Example: For n=4, #IDP_n = 3*9 + 4*8 - 4 + 2 = 27 + 32 - 2 = 57
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MATHEMATICA
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LinearRecurrence[{12, -58, 144, -193, 132, -36}, {2, 5, 17, 57, 185, 593}, 30] (* Harvey P. Dale, Apr 11 2020 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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