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A122367 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). 32
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 (list; graph; refs; listen; history; text; internal format)
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

X-X two connected consecutive "on" intervals (beeeep).

For f(3)=13:

O O O, O O X, O X O, O X X, O X-X, X O O, X O X,

X X O, X-X O, X X X, X X-X, X-X X, X-X-X.

(End)

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

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. 1871-1876 (See Corollary 1 (ii)).

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296

C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.

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), 626-637.

Index entries for linear recurrences with constant coefficients, signature (3,-1).

FORMULA

G.f.: (1-q)/(1-3*q+q^2). More generally, sum( n!/(n-d)!*q^d/prod((1-r*q),r=1..d), d=0..n)/sum( q^d/prod((1-r*q),r=1..d), d=0..n) where n=3.

a(n) = 3*a(n-1)-a(n-2) with a(0) = 1, a(1) = 2.

a(n) = Fibonacci(2n+1) = A000045(2n+1). - Philippe Deléham, Feb 11 2009

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

EXAMPLE

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.

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( d_xi d_xj, i,j = 1..3).

MAPLE

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

a:=n->sum(binomial(n+k, 2*k), k=0..n): seq(a(n), n=0..27); # Zerinvary Lajos, Oct 02 2007

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=0), Z, end_blockRL):Q:=subs([a=Union(ZL3), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n), n=1..28); # Zerinvary Lajos, Mar 08 2008

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((1-x)/(1-3*x+x^2) + O(x^50)) \\ Michel Marcus, Jul 04 2015

CROSSREFS

Cf. A001519, 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

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Last modified April 25 23:17 EDT 2017. Contains 285426 sequences.