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A026097
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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.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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