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A343866
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Number of inequivalent cyclic diagonal Latin squares of order 2n+1 up to rotations, reflections and permutation of symbols.
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2
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1, 0, 1, 1, 0, 2, 3, 0, 4, 4, 0, 5, 3, 0, 7, 7, 0, 2, 9, 0, 10, 10, 0, 11, 7, 0, 13, 4, 0, 14, 15, 0, 6, 16, 0, 17, 18, 0, 8, 19, 0, 20, 8, 0, 22, 10, 0, 8, 24, 0, 25, 25, 0, 26, 27, 0, 28, 10, 0, 14, 22, 0, 13, 31, 0, 32, 16, 0, 34, 34, 0, 20, 14, 0, 37, 37, 0, 14, 39, 0, 20
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OFFSET
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0,6
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COMMENTS
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Also the number of main classes of diagonal Latin squares of order 2n+1 that contain a cyclic Latin square. Compare A341585.
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LINKS
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FORMULA
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a((p-1)/2) = A341585((p-1)/2) for odd prime p.
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EXAMPLE
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a(12) = 3 since there are A123565(25) = 10 cyclic diagonal Latin squares whose first row is in ascending order. Each of these is uniquely defined by the step between rows and form 5 pairs by horizontal or vertical reflection (negating the step between rows). Up to exchanging rows with columns there are 3 distinct classes, so a(12) = 3.
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PROG
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(PARI)
iscanon(n, k, g) = k <= vecmin(g*k%n) && k <= vecmin(g*lift(1/Mod(k, n))%n)
a(n)={if(n==0, 1, my(m=2*n+1); sum(k=1, m-1, gcd(m, k)==1 && gcd(m, k-1)==1 && gcd(m, k+1)==1 && iscanon(m, k, [1, -1])))}
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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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