OFFSET
1,2
COMMENTS
For n >= 3: a(n) = the smallest number > a(n-1) such that ((a(n-2) + a(n-1))*(a(n-2) + a(n))*(a(n-1) + a(n)))/(a(n-2)*a(n-1)*a(n)) is an integer (= 10 for n >= 4).
Number of necklaces with n-1 beads and 3 colors that are the same when turned over and hence have reflection symmetry. Example: For n=4 there are 9 necklaces with the colors A, B and C: AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC. The only necklaces without reflection symmetry are ABC and ACB. - Herbert Kociemba, Nov 24 2016
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3).
FORMULA
a(n) = A038754(n) for n >= 2.
a(2*k) = (3/2)*a(2*k-1) for k >= 2, a(2*k+1) = 2*a(2*k).
G.f.: x*(1 + 3*x + 3*x^2)/(1 - 3*x^2). - Herbert Kociemba, Nov 24 2016
EXAMPLE
For n = 5; a(3) = 6, a(4) = 9, a(5) = 18 before ((6+9)*(6+18)*(9+18)) / (6*9*18) = 10.
MATHEMATICA
Join[{1}, RecurrenceTable[{a[2]==3, a[3]==6, a[n]==3a[n-2]}, a[n], {n, 50}]] (* or *) Transpose[NestList[{#[[2]], #[[3]], 3#[[2]]}&, {1, 3, 6}, 49]][[1]] (* Harvey P. Dale, Oct 19 2011 *)
Rest@ CoefficientList[Series[x (1 + 3 x + 3 x^2)/(1 - 3 x^2), {x, 0, 34}], x] (* Michael De Vlieger, Nov 24 2016 *)
Join[{1}, LinearRecurrence[{0, 3}, {3, 6}, 30]] (* Vincenzo Librandi, Nov 25 2016 *)
PROG
(PARI) x='x+O('x^30); Vec(x*(1+3*x+3*x^2)/(1-3*x^2)) \\ G. C. Greubel, Jan 11 2018
(Magma) I:=[3, 6]; [1] cat [n le 2 select I[n] else 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 11 2018
CROSSREFS
KEYWORD
nonn,easy,less,changed
AUTHOR
Jaroslav Krizek, Nov 27 2010
STATUS
approved