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 A093367 Number of n-bead necklaces using exactly three colors with no adjacent beads having the same color. 2
 0, 0, 2, 3, 6, 11, 18, 33, 58, 105, 186, 349, 630, 1179, 2190, 4113, 7710, 14599, 27594, 52485, 99878, 190743, 364722, 699249, 1342182, 2581425, 4971066, 9587577, 18512790, 35792565, 69273666, 134219793, 260301174, 505294125, 981706830, 1908881897, 3714566310 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Original name: number of periodic cycles of iterative map described by Ma and Wainwright. REFERENCES David W. Hobill and Scott MacDonald (zeened(AT)shaw.ca), Preprint, 2004. P. K-H. Ma and Wainright, A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies, Relativity Today, 1994. LINKS Table of n, a(n) for n=1..37. P. K-H. Ma and Wainright, A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies, Deterministic Chaos in General Relativity, pp. 449-462, 1994. FORMULA a(n) = A000031(n) - (5 + (-1)^n)/2. - Andrew Howroyd, Dec 21 2019 EXAMPLE a(3) = 2 because the two necklaces 123 and 132 have no adjacent equal elements. - Andrew Howroyd, Dec 21 2019 MATHEMATICA Table[Mod[n, 2] - 3 + DivisorSum[n, EulerPhi[n/#] 2^# &]/n, {n, 37}] (* Michael De Vlieger, Dec 22 2019 *) PROG (PARI) a(n)={n%2 - 3 + sumdiv(n, d, eulerphi(n/d)*2^d)/n} \\ Andrew Howroyd, Dec 21 2019 CROSSREFS Column 3 of A330618. Cf. A000031, A093368. Sequence in context: A274621 A291725 A003479 * A054186 A347212 A032156 Adjacent sequences: A093364 A093365 A093366 * A093368 A093369 A093370 KEYWORD nonn AUTHOR N. J. A. Sloane, Apr 28 2004 EXTENSIONS Name changed by Andrew Howroyd, Dec 21 2019 a(1)-a(2) prepended and terms a(20) and beyond from Andrew Howroyd, Dec 21 2019 STATUS approved

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Last modified July 19 22:34 EDT 2024. Contains 374441 sequences. (Running on oeis4.)