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A026097
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4. Also a(n) = sum of numbers in row n+1 of the array T defined in A026082 and a(n) = 24*3^(n-4) for n >= 4.
2
1, 2, 4, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616, 2259436291848
OFFSET
0,2
COMMENTS
Also length of successive strings generated by an alternating Kolakoski (2,4) rule starting at 4 (i.e. string begins with 2 if previous string ends with 4 and vice et versa) : 4-->2222-->44224422-->444422224422444422224422-->... and length of strings are 1,4,8,24,72,... - Benoit Cloitre, Oct 15 2005
Also number of words of length n over alphabet {1,2,3} with no fixed points (a fixed point is value i in position i). - Margaret Archibald, Jun 23 2020
REFERENCES
M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.
LINKS
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
FORMULA
a(n) = 3*a(n-1) for n>3. G.f.: (4*x^3+2*x^2+x-1) / (3*x-1). - Colin Barker, Jun 15 2013
a(n) = floor( (4*n-2)/(n+1) )*a(n-1). Without the floor function the recursion gives the Catalan numbers (A000108). - Hauke Woerpel, Oct 16 2020
MATHEMATICA
CoefficientList[Series[(4 x^3 + 2 x^2 + x - 1)/(3 x - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Join[{1, 2, 4}, NestList[3#&, 8, 30]] (* Harvey P. Dale, May 14 2022 *)
PROG
(PARI) a(n)=if(n>3, 8/27*3^n, 2^n) \\ Charles R Greathouse IV, Jun 23 2020
CROSSREFS
Essentially the same as A005051.
Sequence in context: A114900 A264570 A115115 * A264557 A067646 A152875
KEYWORD
nonn,easy
STATUS
approved