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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). 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
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 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
1-1 two connected consecutive "on" intervals (beeeep).
For f(3)=13:
0 0 0, 0 0 1, 0 1 0, 0 1 1, 0 1-1, 1 0 0, 1 0 1,
1 1 0, 1-1 0, 1 1 1, 1 1-1, 1-1 1, 1-1-1.
(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
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)).
P. Barry and A. Hennessy, The Euler-Seidel 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, 266-296
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
Tanya Khovanova, Recursive Sequences
Diego Marques and Alain Togbé, On the sum of powers of two consecutive Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 10 (2010), 174-176.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
FORMULA
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.
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
a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(k+i-1, k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = A048575(n-1) for n >= 1. - Georg Fischer, Nov 02 2018
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
From J. M. Bergot, May 27 2022: (Start)
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(n-1), 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 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_{j = 1..3} (d_xi d_xj, i).
MAPLE
a:=n->if n=0 then 1; elif n=1 then 2 else 3*a(n-1)-a(n-2); 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
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. similar sequences listed in A238379.
Sequence in context: A001519 A048575 A099496 * A367658 A114299 A112842
KEYWORD
nonn,easy
AUTHOR
Mike Zabrocki, Aug 30 2006
STATUS
approved

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Last modified February 28 03:01 EST 2024. Contains 370379 sequences. (Running on oeis4.)