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A095698 Number of permutations of {1,2,3,...,n} where, for 1 < i <= n, the i-th number has maximized sum of the i-1 absolute differences from all previous numbers of the permutation. 0
1, 2, 4, 6, 14, 18, 46, 54, 146, 162, 454, 486, 1394, 1458, 4246, 4374, 12866, 13122, 38854, 39366, 117074, 118098, 352246, 354294, 1058786, 1062882, 3180454, 3188646, 9549554, 9565938, 28665046, 28697814, 86027906, 86093442, 258149254 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Another variant of A095236: Here each phone after the first selected (which can still be any) is chosen such that the total distance in the normal sense from the chosen phone to all previously-chosen phones in the row is maximized. (Equivalently, the average distance is maximized.) Another space- or privacy-conscious selection strategy. Are there any applications of this sequence to phyllotaxy? Gregarious (or eavesdropping) strategy: If, instead, the total (average) distance is minimized, the sequence generated is 1,2,4,8,16,32,64,128,256,512,..., apparently the nonnegative powers of 2.

In the gregarious case (suggested by the above comment), the permutations that result are exactly those that avoid the permutation patterns 132 and 312. See link to Art of Problem Solving Forums for proof of formula below. - Joel B. Lewis, May 16 2009

LINKS

Table of n, a(n) for n=1..35.

Problem solved on the Art of Problem Solving forum, Urinal-choice permutations. [From Joel B. Lewis, May 16 2009]

FORMULA

a(1) = 1; Conjectured: For k >= 1, a(2k) = a(2k-1) + 2^(k-1) and a(2k+1) = 2*a(2k-1) + a(2k) (needs proof or a reference).

a(2n) = 2 * 3^(n - 1) for n >= 1. a(2n + 1) = 2 * 3^n - 2^n for n >= 0. - Joel B. Lewis, May 16 2009

Conjecture: a(n) = 5*a(n-2)-6*a(n-4). G.f.: x*(1+2*x-x^2-4*x^3)/((1-2*x^2)*(1-3*x^2)). - Colin Barker, Jul 27 2012

Conjecture: a(n) = 2^(((-1)^n + 2*n-5)/4)*((-1)^n-1) - 2*3^(((-1)^n + 2*n-5)/4)*((-1)^n-2). - Luce ETIENNE, Dec 20 2014

EXAMPLE

a(4)=6 as these six permutations of {1,2,3,4} are counted (as in A095236(4)): (1,4,2,3), (1,4,3,2), (2,4,1,3), (3,1,4,2), (4,1,2,3) and (4,1,3,2).

In particular, (2,4,3,1) and (3,1,2,4), counted in A095236(4), are not counted here.

CROSSREFS

Cf. A095236.

Taking every other term gives A008776 (even-indexed terms) and A027649 (odd-indexed terms). - Joel B. Lewis, May 16 2009

Sequence in context: A138307 A323101 A124693 * A277909 A064409 A225078

Adjacent sequences:  A095695 A095696 A095697 * A095699 A095700 A095701

KEYWORD

nonn

AUTHOR

Rick L. Shepherd, Jul 06 2004

EXTENSIONS

More terms from Joel B. Lewis, May 16 2009

STATUS

approved

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)