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A095698
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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.
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0
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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; internal format)
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OFFSET
| 1,2
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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.
Contribution from Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 16 2009: (Start)
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. (End)
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LINKS
| Problem solved on the Art of Problem Solving forum, Urinal-choice permutations. [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 16 2009]
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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. [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 16 2009]
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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.
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CROSSREFS
| Cf. A095236.
Taking every other term gives A008776 (even-indexed terms) and A027649 (odd-indexed terms). [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 16 2009]
Sequence in context: A027712 A138307 A124693 * A064409 A032353 A062112
Adjacent sequences: A095695 A095696 A095697 * A095699 A095700 A095701
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KEYWORD
| nonn
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AUTHOR
| Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 06 2004
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EXTENSIONS
| More terms from Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 16 2009
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