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%I #118 Jul 18 2022 19:22:51
%S 1,2,5,13,34,89,233,610,1597,4181,10946,28657,75025,196418,514229,
%T 1346269,3524578,9227465,24157817,63245986,165580141,433494437,
%U 1134903170,2971215073,7778742049,20365011074,53316291173,139583862445,365435296162,956722026041
%N Dimension of 3-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i != j).
%C Essentially identical to A001519.
%C From _Matthew Lehman_, Jun 14 2008: (Start)
%C Number of monotonic rhythms using n time intervals of equal duration (starting with n=0).
%C Representationally, let 0 be an interval which is "off" (rest),
%C 1 an interval which is "on" (beep),
%C 1 1 two consecutive "on" intervals (beep, beep),
%C 1 0 1 (beep, rest, beep) and
%C 1-1 two connected consecutive "on" intervals (beeeep).
%C For f(3)=13:
%C 0 0 0, 0 0 1, 0 1 0, 0 1 1, 0 1-1, 1 0 0, 1 0 1,
%C 1 1 0, 1-1 0, 1 1 1, 1 1-1, 1-1 1, 1-1-1.
%C (End)
%C Equivalent to the number of one-dimensional graphs of n nodes, subject to the condition that a node is either 'on' or 'off' and that any two neighboring 'on' nodes can be connected. - _Matthew Lehman_, Nov 22 2008
%C Sum_{n>=0} arctan(1/a(n)) = Pi/2. - _Jaume Oliver Lafont_, Feb 27 2009
%H Colin Barker, <a href="/A122367/b122367.txt">Table of n, a(n) for n = 0..1000</a>
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/azarianIJCMS37-40-2012.pdf">Fibonacci Identities as Binomial Sums</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876 (See Corollary 1 (ii)).
%H P. Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry2/barry94r.html">The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences</a>, J. Int. Seq. 13 (2010) # 10.8.2, Example 13.
%H N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="https://arxiv.org/abs/math/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296
%H C. Chevalley, <a href="http://www.jstor.org/stable/2372597">Invariants of finite groups generated by reflections</a>, Amer. J. Math. 77 (1955), 778-782.
%H I. M. Gessel and Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpi.html">Pi and the Fibonacci numbers</a>. - _Jaume Oliver Lafont_, Feb 27 2009
%H Diego Marques and Alain Togbé, <a href="http://dx.doi.org/10.3792/pjaa.86.174">On the sum of powers of two consecutive Fibonacci numbers</a>, Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 10 (2010), 174-176.
%H M. C. Wolf, <a href="http://dx.doi.org/10.1215/S0012-7094-36-00253-3">Symmetric functions of noncommutative elements</a>, Duke Math. J. 2 (1936), 626-637.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F G.f.: (1-q)/(1-3*q+q^2). More generally, (Sum_{d=0..n} (n!/(n-d)!*q^d)/Product_{r=1..d} (1 - r*q)) / (Sum_{d=0..n} q^d/Product_{r=1..d} (1 - r*q)) where n=3.
%F a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 2.
%F a(n) = Fibonacci(2n+1) = A000045(2n+1). - _Philippe Deléham_, Feb 11 2009
%F a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-1+sqrt(5)) + (1+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5). - _Colin Barker_, Oct 14 2015
%F a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(k+i-1, k-i). - _Wesley Ivan Hurt_, Sep 21 2017
%F a(n) = A048575(n-1) for n >= 1. - _Georg Fischer_, Nov 02 2018
%F a(n) = Fibonacci(n)^2 + Fibonacci(n+1)^2. - _Michel Marcus_, Mar 18 2019
%F a(n) = Product_{k=1..n} (1 + 4*cos(2*k*Pi/(2*n+1))^2). - _Seiichi Manyama_, Apr 30 2021
%F From _J. M. Bergot_, May 27 2022: (Start)
%F a(n) = F(n)*L(n+1) + (-1)^n where L(n)=A000032(n) and F(n)=A000045(n).
%F a(n) = (L(n)^2 + L(n)*L(n+2))/5 - (-1)^n.
%F a(n) = 2*(area of a triangle with vertices at (L(n-1), L(n)), (F(n+1), F(n)), (L(n+1), L(n+2))) - 5*(-1)^n for n > 1. (End)
%e a(1) = 2 because x1-x2, x1-x3 are both of degree 1 and are killed by the differential operator d_x1 + d_x2 + d_x3.
%e a(2) = 5 because x1*x2 - x3*x2, x1*x3 - x2*x3, x2*x1 - x3*x1, x1*x1 - x2*x1 - x2*x2 + x1*x2, x1*x1 - x3*x1 - x3*x3 + x3*x1 are all linearly independent and are killed by d_x1 + d_x2 + d_x3, d_x1 d_x1 + d_x2 d_x2 + d_x3 d_x3 and Sum_{j = 1..3} (d_xi d_xj, i).
%p a:=n->if n=0 then 1; elif n=1 then 2 else 3*a(n-1)-a(n-2); fi;
%p A122367List := proc(m) local A, P, n; A := [1,2]; P := [2];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-1]]);
%p A := [op(A), P[-1]] od; A end: A122367List(30); # _Peter Luschny_, Mar 24 2022
%t Table[Fibonacci[2 n + 1], {n, 0, 30}] (* _Vincenzo Librandi_, Jul 04 2015 *)
%o (Magma) [Fibonacci(2*n+1): n in [0..40]]; // _Vincenzo Librandi_, Jul 04 2015
%o (PARI) Vec((1-x)/(1-3*x+x^2) + O(x^50)) \\ _Michel Marcus_, Jul 04 2015
%Y Cf. A001519, A048575, A055105, A055107, A087903, A074664, A008277, A106729, A112340, A122368, A122369, A122370, A122371, A122372.
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 0,2
%A _Mike Zabrocki_, Aug 30 2006
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