0,5

a(n+3) = number of n-bit sequences that avoid 010. Example: For n=4 the 12 sequences are all 4-bit sequences except 0100, 0101, 0010, 1010. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004

Number of different positive braids with n crossings of 3 strands.

This is a_2(n) in the Doroslovacki reference. Note that there is a typo in the paper in the formula for a_2(n): the upper bound in the inner sum should be "n-i" not "i-1". - Max Alekseyev, Jun 26 2007

a(n)=number of peakless Motzkin paths of length n-1 with no UHH...HD's starting at level > 0 (here n>0 and U=(1,1), H=(1,0), D=(1,-1)). Example: a(5)=7 because from all 8 peakless Motzkin paths of length 5 (see A004148) only UUHDD does not qualify. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 13 2004

Conjecture: a(n+1) + |A078065(n)| = 2*A005314(n+1) Formula generated by floretion: + .5'i - .25'j + .25'k + .5i' + .25j' - .25k' - .5'jj' - .5'kk' + .25'ij' - .25'ik' + .25'ji' + .5'jk' - .25'ki' + .5'kj' + e ("jes", "sig" series). Note: as thousands of integer sequences may be associated with any "typical" floretion, the information in parenthesis plays a vital role in looking it back up. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 21 2004

R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.

N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Shifted quasisymmetric functions and the Hopf algebra of peak functions, math.CO/9904105, 1999.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 112.

S. Burckel, Efficient methods for three strand braids (submitted).

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205.

J. Demetrovics et al., On the number of unions in a family of sets, in Combinatorial Math., Proc. 3rd Internat. Conf., Annals NY Acad. Sci., 555 (1989), 150-158.

R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.

T. D. Noe, Table of n, a(n) for n=0..500

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.

R. Doroslovacki, Binary sequences without 011...110 (k-1 1's) for fixed k, Mat. Vesnik 46 (1994), no. 3-4, 93-98.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 98

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

a(n) = 2*a(n-1)-a(n-2)+a(n-3).

a(n) =sum_{j<n}[a(j)]-a(n-2) =A005314(n)-A005314(n-1) =A049853(n-1)-a(n-1) =A005314(n)-a(n-2) - Henry.Bottomley (se16(AT)btinternet.com), Feb 21 2001

a(n+1) = Sum(binomial(n-k, 2k), {k=0...n}). - Richard Ollerton (r.ollerton(AT)uws.edu.au), May 12 2004

G.f.: z(1-z)/(1-2z+z^2-z^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 13 2004

A005251 := proc(n) option remember; if n <= 2 then n elif n = 3 then 4 else 2*A005251(n - 1) - A005251(n - 2) + A005251(n - 3); fi; end;

A005251:=(-1+z)/(-1+2*z-z**2+z**3); [S. Plouffe in his 1992 dissertation.]

Cf. A004148, A049864, A118891.

Bisection of Padovan sequence A000931.

Sequence in context: A003293 A094974 A100671 this_sequence A014167 A103197 A000709

Adjacent sequences: A005248 A005249 A005250 this_sequence A005252 A005253 A005254

nonn,nice,easy

njas, R. K. Guy