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 A236283 The number of orbits of triples of {1,2,...,n} under the action of the dihedral group of order 2n. 5
 1, 4, 5, 10, 13, 20, 25, 34, 41, 52, 61, 74, 85, 100, 113, 130, 145, 164, 181, 202, 221, 244, 265, 290, 313, 340, 365, 394, 421, 452, 481, 514, 545, 580, 613, 650, 685, 724, 761, 802, 841, 884, 925, 970, 1013, 1060, 1105, 1154, 1201, 1252 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In other words, a(n) is the number of equivalence classes of length 3 words with an alphabet of size n where equivalence is up to rotation or reflection of the alphabet. For example when n is 3, the word 112 is equivalent to 223 and 331 by rotation of the alphabet, and these are equivalent to 332, 221 and 113 by reflection of the alphabet. - Andrew Howroyd, Jan 17 2020 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA Conjectures from Colin Barker, Jan 21 2014: (Start) a(n) = (5 + 3*(-1)^n + 2*n^2)/4. a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). G.f.: -x*(2*x^3-3*x^2+2*x+1) / ((x-1)^3*(x+1)). (End) From Andrew Howroyd, Jan 17 2020: (Start) The above conjectures are true and can be derived from the following formulas for even and odd n. a(n) = (n-2)*(n + 2)/2 + 4 for even n. a(n) = (n-1)*(n + 1)/2 + 1 for odd n. (End) a(n) = A081352(n - 1) - A116940(n - 1). - Miko Labalan, Nov 12 2016 EXAMPLE For n = 3 there are 5 orbits of triples: [[1,1,1], [2,2,2], [3,3,3]], [[1,1,2], [2,2,3], [1,1,3], [3,3,1], [3,3,2], [2,2,1]], [[1,2,1], [2,3,2], [1,3,1], [3,1,3], [3,2,3], [2,1,2]], [[1,2,2], [2,3,3], [1,3,3], [3,1,1], [3,2,2], [2,1,1]], [[1,2,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1], [2,1,3]]. PROG (GAP) a:=function(n) local g, orbs; g:=DihedralGroup(IsPermGroup, 2*n); orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 3), OnTuples ); return Size(orbs); end;; (PARI) a(n) = {(5 + 3*(-1)^n + 2*n^2)/4} \\ Andrew Howroyd, Jan 17 2020 CROSSREFS Cf. A236332 (4-tuples). Sequence in context: A094415 A114517 A283246 * A322610 A322468 A116930 Adjacent sequences:  A236280 A236281 A236282 * A236284 A236285 A236286 KEYWORD nonn,easy AUTHOR W. Edwin Clark, Jan 21 2014 STATUS approved

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Last modified May 24 19:12 EDT 2020. Contains 334580 sequences. (Running on oeis4.)