A122367 Dimension of 3-variable non-commutative harmonics (twisted derivative) of order n. The dimension of the space of non-commutative polynomials of degree n 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).

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

Offset: 0

Author

Mike Zabrocki, Aug 30 2006

Keywords

• nonn
• easy

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

This is the limit sequence for certain generalized Pell numbers. - Gregory L. Simay, Oct 21 2024

Examples

			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).
		

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)).
• Paul 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
• Ron Knott, Pi and the Fibonacci numbers. - _Jaume Oliver Lafont_, Feb 27 2009
• 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.
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.
• 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).

Crossrefs

Cf. A001519, A048575, A055105, A055107, A087903, A074664, A008277, A106729, A112340, A122368, A122369, A122370, A122371, A122372.

Cf. similar sequences listed in A238379.

Programs

Magma

[Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 04 2015

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 *)

PARI

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

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)

G.f.: (1-x)/(1-3x+x^2) = 1/(1-2x-x^2-x^3-x^4-...) - Gregory L. Simay, Oct 21 2024

E.g.f.: exp(3*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Nov 07 2024

From Peter Bala, May 04 2025: (Start)

a(n) = sqrt(2/5) * sqrt( 1 - T(2*n+1, - 3/2) ), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind.

a(2*n+1/2) = sqrt(5)*a(n)^2 - 2/sqrt(5).

a(3*n+1) = 5*a(n)^3 - 3*a(n); hence a(n) divides a(3*n+1).

a(4*n+3/2) = 5^(3/2)*a(n)^4 - 4*sqrt(5)*a(n)^2 + 2/sqrt(5).

a(5*n+2) = (5^2)*a(n)^5 - 5*5*a(n)^3 + 5*a(n); hence a(n) divides a(5*n+2).

See A034807 for the unsigned coefficients [1, 2; 1, 3; 1, 4, 2; 1, 5, 5; ...].

In general, for k >= 0, a(k*n + (k-1)/2) = a(-1/2) * T(k, a(n)/a(-1/2)), where a(n) = (2^(-1-n)*((3 - sqrt(5))^n *(-1 + sqrt(5)) + (1 + sqrt(5))*(3 + sqrt(5))^n)) / sqrt(5), as given above, and a(-1/2) = 2/sqrt(5).

The aerated sequence [b(n)]n>=1 = [1, 0, 2, 0, 5, 0, 13, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -5, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy.

Sum_{n >= 1} 1/(a(n) - 1/a(n)) = 1 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A001906(n) - 1/A001906(n+1).) (End)