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%I #13 Sep 08 2022 08:46:22
%S 1,5,18,56,161,443,1190,3156,8315,21835,57246,149970,392743,1028351,
%T 2692416,7049018,18454775,48315461,126491780,331160070,866988641,
%U 2269806085,5942429868,15557483796,40730021821,106632581993,279167724510,730870591916,1913444051645,5009461563455
%N 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.
%H Ramón Flores, Juan González-Meneses, <a href="https://arxiv.org/abs/1808.02755">On lexicographic representatives in braid monoids</a>, arXiv:1808.02755 [math.GR], 2018.
%H Volker Gebhardt, Juan González-Meneses, <a href="https://doi.org/10.1016/j.jcta.2012.07.003">Generating random braids</a>, J. Comb. Th. A 120 (1), 2013, 111-128.
%F a(n) = Sum_{i=1..n} (binomial(n+1-i, 2)+1)*Fibonacci(2*i).
%F 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
%t Table[Sum[(Binomial[n + 1 - k, 2] + 1) Fibonacci[2 k], {k, n}], {n, 30}] (* _Vincenzo Librandi_, Aug 09 2018 *)
%o (PARI) a(n) = sum(i=1, n, (binomial(n+1-i, 2)+1)*fibonacci(2*i));
%o (Magma) [&+[(Binomial(n+1-k, 2)+1)*Fibonacci(2*k): k in [1..n]]: n in [1..30]]; // _Vincenzo Librandi_, Aug 09 2018
%o (GAP) List([1..30],n->Sum([1..n],i->(Binomial(n+1-i,2)+1)*Fibonacci(2*i))); # _Muniru A Asiru_, Aug 09 2018
%K nonn
%O 1,2
%A _Michel Marcus_, Aug 09 2018
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