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.

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

Author

Clark Kimberling

Keywords

• nonn
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
• Index entries for linear recurrences with constant coefficients, signature (3).

Crossrefs

Essentially the same as A005051.

Programs

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 *)

PARI

a(n)=if(n>3,8/27*3^n,2^n) \\ Charles R Greathouse IV, Jun 23 2020

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