This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A183155 #47 Mar 08 2025 12:32:04
%S A183155 1,1,3,9,23,53,115,241,495,1005,2027,4073,8167,16357,32739,65505,
%T A183155 131039,262109,524251,1048537,2097111,4194261,8388563,16777169,
%U A183155 33554383,67108813,134217675,268435401,536870855,1073741765
%N A183155 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).
%C A183155 a(n) is also the number of dominating sets in the (n+1)-path complement graph. - _Eric W. Weisstein_, Apr 11 2018
%H A183155 Zachary Hamaker, Eric Marberg, and Brendan Pawlowski, <a href="https://arxiv.org/abs/1706.06665">Fixed-point-free involutions and Schur P-positivity</a>, arXiv:1706.06665 [math.CO], 2017.
%H A183155 R. Kehinde and A. Umar, <a href="http://arxiv.org/abs/1101.0049">On the semigroup of partial isometries of a finite chain</a>, arXiv:1101.0049 [math.GR], 2010.
%H A183155 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>.
%H A183155 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>.
%H A183155 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A183155 a(n) = A183154(n,0).
%F A183155 a(n) = 2^(n+1) - (2*n+1).
%F A183155 a(0)=1; for n > 0, a(n) = 2*a(n-1) + 2*n - 3. - _Vincenzo Librandi_, Feb 05 2011
%F A183155 G.f.: (-1+3*x-4*x^2)/((2*x-1)*(x-1)^2). - _R. J. Mathar_, Feb 06 2011
%F A183155 From _Elmo R. Oliveira_, Mar 07 2025: (Start)
%F A183155 E.g.f.: exp(x)*(2*exp(x) - (1 + 2*x)).
%F A183155 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). (End)
%e A183155 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.
%t A183155 Table[1 + 2^(1 + n) - 2 (1 + n), {n, 0, 20}] (* or *)
%t A183155 LinearRecurrence[{4, -5, 2}, {1, 3, 9}, {0, 20}] (* or *)
%t A183155 CoefficientList[Series[(-1 + 3 x - 4 x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Apr 11 2018 *)
%o A183155 (PARI) a(n) = 2^(n+1)-(2*n+1); \\ _Altug Alkan_, Apr 12 2018
%Y A183155 Cf. A000918 (first differences), A183154.
%K A183155 nonn,easy
%O A183155 0,3
%A A183155 _Abdullahi Umar_, Dec 28 2010