login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A093387 2^(n-1) - binomial(n, floor(n/2)). 8
0, 0, 1, 2, 6, 12, 29, 58, 130, 260, 562, 1124, 2380, 4760, 9949, 19898, 41226, 82452, 169766, 339532, 695860, 1391720, 2842226, 5684452, 11576916, 23153832, 47050564, 94101128, 190876696, 381753392, 773201629, 1546403258, 3128164186, 6256328372, 12642301534 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Suppose n >= 3. Let e_1,...,e_n be n unit-vectors which generate Euclidean space R_n and let l_n = {x= sum a_i e_i | a_1 >= a_2 >= ... >= a_n >= 0 }. Consider the hypercube H_n with vertices h_1,...,h_{2^n} = {epsilon_1 e_1+...+ epsilon_n e_n}.

For each element x in l_n we build 2^n "statements" by taking the inner product of x with h_i. We call a statement true if (x,h_i)>0 and false if (x,h_i)<0. Two vectors x and y are indistinguishable if all statements produced by x and y are equal.

For each set of indistinguishable vectors we chose one vector, which is called the representative. The sequence gives the number of representatives.

Hankel transform is A127365. - Paul Barry, Jan 11 2007

Number of up-steps starting at level 0 in all dispersed Dyck paths of length n-1 (that is, in Motzkin paths of length n-1 with no (1,0)-steps at positive heights). - Emeric Deutsch, May 30 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Matthijs Coster, Sequences

Matthijs Coster, Statements and Representatives, 2004.

FORMULA

a(n) = A000079(n-1)-A001405(n).

a(n+1) = sum{k=2..n, C(n,floor((n-k)/2)}; - Paul Barry, Jan 11 2007

a(2n) = 2*a(2n-1). - Emeric Deutsch, May 30 2011

a(n+1) = sum_{k>=0} k*A191310(n,k). - Emeric Deutsch, May 30 2011

G.f.: (1-sqrt(1-4*z^2))^2/(4*z*(1-2*z)). - Emeric Deutsch, May 30 2011

Conjecture: (n+1)*a(n) +2*(-n-1)*a(n-1) +4*(-n+2)*a(n-2) +8*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012

EXAMPLE

a(5)=6 because, denoting U=(1,1), D=(1,-1), H=(1,0), in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+1+1+1+2+1=6 U steps starting at level 0. - Emeric Deutsch, May 30 2011

MAPLE

A093387:=n->2^(n-1)-binomial(n, floor(n/2)); seq(A093387(n), n=1..50); # Wesley Ivan Hurt, Dec 01 2013

MATHEMATICA

Table[2^(n - 1) - Binomial[n, Floor[n/2]], {n, 50}] (* Wesley Ivan Hurt, Dec 01 2013 *)

PROG

(PARI) a(n) = 2^(n-1) - binomial(n, n\2); \\ Michel Marcus, Aug 13 2013

CROSSREFS

Sequence in context: A183467 A057582 A094779 * A229487 A195166 A225646

Adjacent sequences:  A093384 A093385 A093386 * A093388 A093389 A093390

KEYWORD

nonn

AUTHOR

Matthijs Coster, Apr 29 2004

EXTENSIONS

Offset corrected, R. J. Mathar, Jun 04 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 23 01:23 EST 2014. Contains 249836 sequences.