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 A174075 Number of circular permutations of length n without modular consecutive triples i,i+2,i+4. 6
 1, 6, 18, 93, 600, 4320, 35168, 321630, 3257109, 36199458, 438126986, 5736774869, 80808984725, 1218563192160, 19587031966352, 334329804180135, 6039535339644630, 115118210695441900, 2308967760171049528, 48613722701440862328, 1072008447320752890459 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 COMMENTS Circular permutations are permutations whose indices are from the ring of integers modulo n. REFERENCES Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. LINKS Table of n, a(n) for n=3..23. FORMULA a(n) = A165962(n) for odd n. EXAMPLE Since a(5)=18, there are (5-1)!-18=4 circular permutations with modular consecutive triples i,i+2,i+4 in all circular permutations of length 5. These are exactly (0,2,4,1,3), (0,2,4,3,1), (0,4,2,1,3), and (0,3,2,4,1). Note some have more than one modular progression. MATHEMATICA f[i_, n_, k_]:=If[i==0 && k==0, 1, If[i==n && n==k, 1, Binomial[k-1, k-i]*Binomial[n-k-1, k-i-1] + 2*Binomial[k-1, k-i-1]*Binomial[n-k-1, k-i-1]+Binomial[k-1, k-i-1]*Binomial[n-k-1, k-i]]]; w1[i_, n_, k_]:=If[n-2k+i<0, 0, If[n-2k+i==0, 1, (n-2k+i-1)!]]; a[n_, k_]:=Sum[f[i, n, k]*w1[i, n, k], {i, 0, k}]; A165962[n_]:=(n-1)!+Sum[(-1)^k*a[n, k], {k, 1, n}]; b[n_, k_]:=Sum[Sum[Sum[f[j, n/2, p]*f[i-j, n/2, k-p]*w2[i, j, n, k, p], {p, 0, k}], {j, 0, i}], {i, 0, k-1}]; w2[i_, j_, n_, k_, p_]:=If[n/2-2p+j<=0 || n/2-2(k-p)+(i-j)<=0, 0, (n-2k+i-1)!]; A216727[n_?EvenQ]:=(n-1)!+Sum[(-1)^k*b[n, k], {k, 1, n}]; A216727[n_?OddQ]:=A165962[n]; Table[A216727[n], {n, 3, 23}] (* David Scambler, Sep 18 2012 *) CROSSREFS Column 1 of A216726. Cf. A165962, A174072, A174073, A174074. Sequence in context: A194995 A104970 A216727 * A151470 A280096 A009573 Adjacent sequences: A174072 A174073 A174074 * A174076 A174077 A174078 KEYWORD nonn AUTHOR Isaac Lambert, Mar 06 2010 STATUS approved

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Last modified November 28 08:58 EST 2023. Contains 367411 sequences. (Running on oeis4.)