A317849 Number of states of the Finite State Automaton Gn accepting the language of maximal (or minimal) lexicographic representatives of elements in the positive braid monoid An.

1, 5, 18, 56, 161, 443, 1190, 3156, 8315, 21835, 57246, 149970, 392743, 1028351, 2692416, 7049018, 18454775, 48315461, 126491780, 331160070, 866988641, 2269806085, 5942429868, 15557483796, 40730021821, 106632581993, 279167724510, 730870591916, 1913444051645, 5009461563455

Offset: 1

Author

Michel Marcus, Aug 09 2018

Keywords

• nonn

Links

• Ramón Flores, Juan González-Meneses, On lexicographic representatives in braid monoids, arXiv:1808.02755 [math.GR], 2018.
• Volker Gebhardt, Juan González-Meneses, Generating random braids, J. Comb. Th. A 120 (1), 2013, 111-128.

Programs

GAP

List([1..30],n->Sum([1..n],i->(Binomial(n+1-i,2)+1)*Fibonacci(2*i))); # Muniru A Asiru, Aug 09 2018

Magma

[&+[(Binomial(n+1-k, 2)+1)*Fibonacci(2*k): k in [1..n]]: n in [1..30]]; // Vincenzo Librandi, Aug 09 2018

Mathematica

Table[Sum[(Binomial[n + 1 - k, 2] + 1) Fibonacci[2 k], {k, n}], {n, 30}] (* Vincenzo Librandi, Aug 09 2018 *)

PARI

a(n) = sum(i=1, n, (binomial(n+1-i, 2)+1)*fibonacci(2*i));

Formula

a(n) = Sum_{i=1..n} (binomial(n+1-i, 2)+1)*Fibonacci(2*i).

Conjecture: g.f. -x*(1-x+x^2) / ( (x^2-3*x+1)*(x-1)^3 ). a(n) = 2*A001519(n+1) -n*(n+1)/2 -2 = 2*A001519(n+1)-A152948(n+2). - R. J. Mathar, Aug 17 2018