2,2

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006.

Comment from Dmitry Kamenetsky (Dmitry.Kamenetsky(AT)rsise.anu.edu.au), Nov 29 2006: This is the length of the longest non self-intersecting spiral drawn on an N X N grid. E.g. for n=5 the spiral has length 17:

10111

10101

10101

10001

11111

Graham Cormode, Notes on the organization number of a permutation.

a(2)=1; for n>2, a(n)=a(n-1)+n-1+(n-1 mod 2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 12 2003

T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2) where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006.

Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2007

Table[ Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)

(PARI) a(n)=if(n<2, 0, n^2\2-1)

Complement of A047839. Also A007590 - 1. First differences give A052928.

Cf. A135151, A135152.

Sequence in context: A081515 A023234 A134707 this_sequence A029715 A088803 A088206

Adjacent sequences: A047835 A047836 A047837 this_sequence A047839 A047840 A047841

nonn

Michael Somos