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A321672
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Number of chiral pairs of rows of length 5 using up to n colors.
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1
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0, 0, 12, 108, 480, 1500, 3780, 8232, 16128, 29160, 49500, 79860, 123552, 184548, 267540, 378000, 522240, 707472, 941868, 1234620, 1596000, 2037420, 2571492, 3212088, 3974400, 4875000, 5931900, 7164612, 8594208, 10243380, 12136500
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (n^5 - n^3) / 2.
G.f.: (Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1) - Sum_{j=1..3} S2(3,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x * Sum_{k=1..4} A145883(5,k) * x^k / (1-x)^6.
E.g.f.: (Sum_{k=1..5} S2(5,k)*x^k - Sum_{k=1..3} S2(3,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>5, a(n) = Sum_{j=1..6} -binomial(j-7,j) * a(n-j).
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EXAMPLE
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For a(0)=0 and a(1)=0, there are no chiral rows using fewer than two colors. For a(2)=12, the chiral pairs are AAAAB-BAAAA, AAABA-ABAAA, AAABB-BBAAA, AABAB-BABAA, AABBA-ABBAA, AABBB-BBBAAA, ABAAB-BAABA, ABABB-BBABA, ABBAB-BABBA, ABBBB-BBBBA, BAABB-BBAAB, and BABBB-BBBAB.
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MATHEMATICA
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Table[(n^5-n^3)/2, {n, 0, 40}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 12, 108, 480, 1500}, 40]
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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