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

1,2

Links

Table of n, a(n) for n=1..30.

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.

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

Mathematica

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

Prog

(PARI) a(n) = sum(i=1, n, (binomial(n+1-i, 2)+1)*fibonacci(2*i));
(Magma) [&+[(Binomial(n+1-k, 2)+1)*Fibonacci(2*k): k in [1..n]]: n in [1..30]]; // Vincenzo Librandi, Aug 09 2018
(GAP) List([1..30], n->Sum([1..n], i->(Binomial(n+1-i, 2)+1)*Fibonacci(2*i))); # Muniru A Asiru, Aug 09 2018

Crossrefs

Sequence in context: A145129 A001793 A325919 * A307572 A325923 A387263

Adjacent sequences: A317846 A317847 A317848 * A317850 A317851 A317852

Keyword

nonn

Author

Michel Marcus, Aug 09 2018

Status

approved