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A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

%I A065167 #5 May 02 2017 22:17:15
%S A065167 1,2,2,3,4,4,4,1,6,6,5,6,2,8,8,6,3,8,4,10,10,7,8,1,10,6,12,12,8,5,10,
%T A065167 2,12,8,14,14,9,10,3,12,4,14,10,16,16,10,7,12,1,14,6,16,12,18,18,11,
%U A065167 12,5,14,2,16,8,18,14,20,20,12,9,14,3,16,4,18,10,20,16,22,22,13,14,7,16,1
%N A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).
%C A065167 Simple periodic site swap permutations of natural numbers.
%C A065167 Row n of the table (starting from n=0) gives a permutation of natural numbers corresponding to the simple, infinite, periodic site swap pattern ...nnnnn...
%H A065167 Joe Buhler and R. L. Graham, <a href="http://www.cecm.sfu.ca/organics/papers/buhler/index.html">Juggling Drops and Descents</a>, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
%H A065167 Juggling Information Service, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/">Site Swap FAQs</a>
%F A065167 Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then the n-th term of the k-th row is f(g(n)+k).
%e A065167 Table begins:
%e A065167 1 2 3 4 5 6 7 ...
%e A065167 2 4 1 6 3 8 5 ...
%e A065167 4 6 2 8 1 10 3 ...
%e A065167 6 8 4 10 2 12 1 ...
%p A065167 PerSS_table := (n) -> PerSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2))); PerSS := (n,c) -> Z2N(N2Z(n)+c);
%p A065167 N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
%p A065167 [seq(PerSS_table(j),j=0..119)];
%Y A065167 Successive rows and associated site swap sequences, starting from the zeroth row: (A000027, A000004), (A065164, A000012), (A065165, A007395), (A065166, A010701). Cf. also A065171, A065174, A065177. trinv given at A054425.
%K A065167 nonn,tabl
%O A065167 0,2
%A A065167 _Antti Karttunen_, Oct 19 2001