

A122367


Dimension of 3variable noncommutative harmonics (twisted derivative). The dimension of the space of noncommutative 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).


33



1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041
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OFFSET

0,2


COMMENTS

Essentially identical to A001519.
From Matthew Lehman, Jun 14 2008: (Start)
Number of monotonic rhythms using n time intervals of equal duration (starting with n=0).
Representationally, let O be an interval which is "off" (rest),
X an interval which is "on" (beep),
X X two consecutive "on" intervals (beep, beep),
X O X (beep, rest, beep) and
XX two connected consecutive "on" intervals (beeeep).
For f(3)=13:
O O O, O O X, O X O, O X X, O XX, X O O, X O X,
X X O, XX O, X X X, X XX, XX X, XXX.
(End)
Equivalent to the number of onedimensional 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
Sum_{n>=0} arctan(1/a(n)) = Pi/2.  Jaume Oliver Lafont, Feb 27 2009


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 18711876 (See Corollary 1 (ii)).
P. Barry, A. Hennessy, The EulerSeidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 13
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266296
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778782.
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
Tanya Khovanova, Recursive Sequences
Ron Knott, Pi and the Fibonacci numbers.  Jaume Oliver Lafont, Feb 27 2009
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626637.
Index entries for linear recurrences with constant coefficients, signature (3,1).


FORMULA

G.f.: (1q)/(13*q+q^2). More generally, (Sum_{d=0..n} (n!/(nd)!*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.
a(n) = 3*a(n1)  a(n2) with a(0) = 1, a(1) = 2.
a(n) = Fibonacci(2n+1) = A000045(2n+1).  Philippe Deléham, Feb 11 2009
a(n) = (2^(1n)*((3sqrt(5))^n*(1+sqrt(5)) + (1+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5).  Colin Barker, Oct 14 2015
a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(k+i1, ki).  Wesley Ivan Hurt, Sep 21 2017
a(n) = A048575(n1) for n >= 1.  Georg Fischer, Nov 02 2018


EXAMPLE

a(1) = 2 because x1x2, x1x3 are both of degree 1 and are killed by the differential operator d_x1 + d_x2 + d_x3.
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).


MAPLE

a:=n>if n=0 then 1; elif n=1 then 2 else 3*a(n1)a(n2); fi;


MATHEMATICA

Table[Fibonacci[2 n + 1], {n, 0, 30}] (* Vincenzo Librandi, Jul 04 2015 *)


PROG

(MAGMA) [Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 04 2015
(PARI) Vec((1x)/(13*x+x^2) + O(x^50)) \\ Michel Marcus, Jul 04 2015


CROSSREFS

Cf. A001519, A048575, A055105, A055107, A087903, A074664, A008277, A112340, A122368, A122369, A122370, A122371, A122372.
Cf. similar sequences listed in A238379.
Sequence in context: A001519 A048575 A099496 * A114299 A112842 A097417
Adjacent sequences: A122364 A122365 A122366 * A122368 A122369 A122370


KEYWORD

nonn,easy


AUTHOR

Mike Zabrocki, Aug 30 2006


STATUS

approved



