

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).


35



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

Number of monotonic rhythms using n time intervals of equal duration (starting with n=0).
Representationally, let 0 be an interval which is "off" (rest),
1 an interval which is "on" (beep),
1 1 two consecutive "on" intervals (beep, beep),
1 0 1 (beep, rest, beep) and
11 two connected consecutive "on" intervals (beeeep).
For f(3)=13:
0 0 0, 0 0 1, 0 1 0, 0 1 1, 0 11, 1 0 0, 1 0 1,
1 1 0, 11 0, 1 1 1, 1 11, 11 1, 111.
(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


LINKS



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) = (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) = Fibonacci(n)^2 + Fibonacci(n+1)^2.  Michel Marcus, Mar 18 2019
a(n) = Product_{k=1..n} (1 + 4*cos(2*k*Pi/(2*n+1))^2).  Seiichi Manyama, Apr 30 2021
a(n) = F(n)*L(n+1) + (1)^n where L(n)=A000032(n) and F(n)=A000045(n).
a(n) = (L(n)^2 + L(n)*L(n+2))/5  (1)^n.
a(n) = 2*(area of a triangle with vertices at (L(n1), L(n)), (F(n+1), F(n)), (L(n+1), L(n+2)))  5*(1)^n for n > 1. (End)


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;
A122367List := proc(m) local A, P, n; A := [1, 2]; P := [2];
for n from 1 to m  2 do P := ListTools:PartialSums([op(A), P[1]]);
A := [op(A), P[1]] od; A end: A122367List(30); # Peter Luschny, Mar 24 2022


MATHEMATICA



PROG

(PARI) Vec((1x)/(13*x+x^2) + O(x^50)) \\ Michel Marcus, Jul 04 2015


CROSSREFS

Cf. A001519, A048575, A055105, A055107, A087903, A074664, A008277, A106729, A112340, A122368, A122369, A122370, A122371, A122372.
Cf. similar sequences listed in A238379.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



