%I A132101
%S A132101 1,1,3,11,65,513,5363,68219,1016481,17243105,327431363
%N A132101 Number of distinct Tsuro tiles which are digonal in shape and have n 
               points per side.
%C A132101 Turning over is not allowed.
%C A132101 See A132100 for definition and comments.
%C A132101 Comments from Ross Drewe (rd(AT)labyrinth.net.au), Mar 16 2008: (Start) 
               This is the number of arrangements of n pairs which are equivalent 
               under the joint operation of sequence reversal and permutations of 
               labels. Assume that the elements of n distinct pairs are labelled 
               to show the pair of origin, eg, [1 1], [2 2]. The number of distinguishable 
               ways of arranging these elements falls as the conditions are made 
               more general:
%C A132101 a(n) = A000680: element order is significant and the labels are distinguishable;
%C A132101 b(n) = A001147: element order is significant but labels are not distinguishable, 
               i.e. all label permutations of a given sequence are equivalent;
%C A132101 c(n) = A132101: element order is weakened (reversal allowed) and all 
               label permutations are equivalent;
%C A132101 d(n) = A047974: reversal allowed, all label permutations are equivalent 
               and equivalence class maps to itself under joint operation.
%C A132101 Those classes that do not map to themselves form reciprocal pairs of 
               classes under the joint operation and their number is r(n). Then 
               c = b - r/2 = b - (b - d)/2 = (b+d)/2. A formula for r(n) is not 
               available, but formulae are available for b(n) = A001147 and d(n) 
               = A047974, allowing an explicit formula for this sequence.
%C A132101 c(n) is useful in extracting structure information without regard to 
               pair ordering (see example). c(n) terms also appear in formulae related 
               to binary operators, eg, the number of binary operators in a k-valued 
               logic that are invertible in 1 operation.
%C A132101 a(n) = (b(n) + c(n))/2, where b(n) = (2n)!/(2^n * n!), c(n) = sum(k=0,
               1,...floor(n/2)) (n!/((n-2*k)! * k!)
%C A132101 For 3 pairs, the arrangement A = [112323] is the same as B = [212133] 
               under the permutation of the labels [123] -> [312] plus reversal 
               of the elements, or vice versa. The unique structure common to A 
               and B is {1 intact pair + 2 interleaved pairs}, where the order is 
               not significant (contrast A001147). (End)
%Y A132101 Cf. A132100-A132105, A007769, A001147, A054499.
%Y A132101 Cf. A000680, A001147, A047974.
%Y A132101 Sequence in context: A069725 A096655 A030226 this_sequence A077428 A074504 
               A126115
%Y A132101 Adjacent sequences: A132098 A132099 A132100 this_sequence A132102 A132103 
               A132104
%K A132101 nonn
%O A132101 0,3
%A A132101 Keith F. Lynch (kfl(AT)KeithLynch.net), Oct 31 2007
%E A132101 3 more terms from Ross Drewe (rd(AT)labyrinth.net.au), Mar 16 2008