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A183155
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The number of order-preserving partial isometries (of an n-chain) of fix zero (fix of alpha = 0). Equivalently, the number of order-preserving partial derangement isometries (of an n-chain).
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6
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1, 1, 3, 9, 23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, 65505, 131039, 262109, 524251, 1048537, 2097111, 4194261, 8388563, 16777169, 33554383, 67108813, 134217675, 268435401, 536870855, 1073741765
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of dominating sets in the (n+1)-path complement graph. - Eric W. Weisstein, Apr 11 2018
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LINKS
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FORMULA
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a(n) = 2^(n+1) - (2*n+1).
G.f. ( -1+3*x-4*x^2 ) / ( (2*x-1)*(x-1)^2 ). - R. J. Mathar, Feb 06 2011
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EXAMPLE
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a(3) = 9 because there are exactly 9 order-preserving partial derangement isometries (on a 3-chain) , namely: empty map; 1-->2; 1-->3; 2-->1; 2-->3; 3-->1; 3-->2; (1,2)-->(2,3); (2,3)-->(1,2) - the mappings are coordinate-wise.
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MATHEMATICA
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Table[1 + 2^(1 + n) - 2 (1 + n), {n, 0, 20}] (* or *)
LinearRecurrence[{4, -5, 2}, {1, 3, 9}, {0, 20}] (* or *)
CoefficientList[Series[(-1 + 3 x - 4 x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)
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PROG
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(PARI) a(n) = 2^(n+1)-(2*n+1); \\ Altug Alkan, Apr 12 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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