0,2

Essentially identical to A001519

Comments from Matthew Lehman (matt.comicopia(AT)gmail.com), Jun 14 2008: 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

Contribution from Matthew Lehman (matt.comicopia(AT)gmail.com), Nov 22 2008: (Start)

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. (End)

Sum_{n>=0} atan(1/a(n)) = Pi/2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 27 2009]

N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, to appear Canad. J. Math., arXiv:math.CO/0502082

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

M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.

Tanya Khovanova, Recursive Sequences

R. Knott, Pi and the Fibonacci numbers [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 27 2009]

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). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

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 lin. ind. 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)

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 (zerinvarylajos(AT)yahoo.com), 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 (zerinvarylajos(AT)yahoo.com), Mar 08 2008

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

Sequence in context: A027933 A141448 A011783 this_sequence A001519 A048575 A099496

Adjacent sequences: A122364 A122365 A122366 this_sequence A122368 A122369 A122370

nonn

Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006