login
A084084
Length of lists created by n substitutions k -> Range[0,1+Mod[k+1,3]] starting with {0}.
1
1, 3, 9, 28, 86, 265, 816, 2513, 7739, 23833, 73396, 226030, 696081, 2143648, 6601569, 20330163, 62608681, 192809420, 593775046, 1828587033, 5631308624, 17342153393, 53406819691, 164471408185, 506505428836, 1559831901918
OFFSET
0,2
COMMENTS
Transformation invert T109 gave a match with A078039; T100 binomial gave a match with A012781; equivalent to replacements 0 -> {0,1,2}; 1 -> {0,1,2,3}; 2 -> {0,1}, 3 -> {0,1,2} operating n times with {0}.
LINKS
Tomislav Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.
FORMULA
G.f.: (1+x)/(1-2*x-3*x^2-x^3).
a(n) = A000931(4*n + 6). - Michael Somos, Sep 18 2012
EXAMPLE
{0}, {0,1,2}, {0,1,2,0,1,2,3,0,1}, {0,1,2,0,1,2,3,0,1,0,1,2,0,1,2,3,0,1,0,1,2,0,1,2,0,1,2,3} have lengths 1, 3, 9, 28.
G.f. = 1 + 3*x + 9*x^2 + 28*x^3 + 86*x^4 + 265*x^5 + 816*x^6 + ...
MATHEMATICA
Length/@Flatten/@NestList[ # /. k_Integer:>Range[0, 1+Mod[k+1, 3]]&, {0}, 8]
LinearRecurrence[{2, 3, 1}, {1, 3, 9}, 41] (* G. C. Greubel, Oct 16 2022 *)
PROG
(Magma) [n le 3 select 3^(n-1) else 2*Self(n-1) +3*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Oct 16 2022
(SageMath)
def A084084_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)/(1-2*x-3*x^2-x^3) ).list()
A084084_list(40) # G. C. Greubel, Oct 16 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, May 11 2003
STATUS
approved