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.
a(10) = 105 = A000055(10) - 1 because all trees with 10 vertices are lobsters except this one:
o-o-o
/
o-o-o-o
\
o-o-o
Also, all trees with 10 vertices are linear (all vertices of degree >2 belong to a single path) except this one:
o o
\ /
o
|
o
/ \
o o
/ \ / \
o o o o
eta = QPochhammer;
s[n_] := With[{ox = O[x]^n}, x^2 ((1/eta[x + ox] - 1/(1 - x))^2/(1 - x/eta[x + ox]) + (1/eta[x^2 + ox] - 1/(1 - x^2))(1 + x/eta[x + ox])/(1 - x^2/eta[x^2 + ox]))/2 + x/eta[x + ox] - x^3/((1 - x)^2*(1 + x))];
CoefficientList[s[32], x] // Rest (* Jean-François Alcover, Nov 17 2020, after Andrew Howroyd *)
s(n)={my(ox=O(x^n)); x^2*((1/eta(x+ox)-1/(1-x))^2/(1-x/eta(x+ox)) + (1/eta(x^2+ox)-1/(1-x^2))*(1+x/eta(x+ox))/(1-x^2/eta(x^2+ox)))/2 + x/eta(x+ox) - x^3/((1-x)^2*(1+x))}
Vec(s(30)) \\ Andrew Howroyd, Nov 02 2017
For n = 1, there is only one room to search, so a(1) = 1. For n = 2, the prince searches room 1 on the first night. If the princess is not there that means she was in room 2. If the prince searches room 1 again then he is guaranteed to see the princess as she has to move from room 2 to room 1 (she cannot stay in the same room). So a(2) = 2. For n = 3, the prince searches room 2 on the first night. If the princess is not there that means she was either in room 1 or 3. On the second night she must go to room 2 and this is where the prince will find her. So a(3) = 2. For n = 4, a solution that guarantees to find the princess in a(4)=4 nights is to search rooms (2,3,3,2). For n = 5, a solution that guarantees to find the princess in a(5)=6 nights is to search rooms (2,3,4,4,3,2). In the general case for n >= 3, a solution guaranteeing success in the minimum number of nights is to search rooms (2,3,...,n-1,n-1,...,3,2), so a(n) = 2*n - 4.
CoefficientList[ Series[(2x^3 - x^2 + 1)/(x - 1)^2, {x, 0, 62}], x] (* Robert G. Wilson v, Mar 12 2018 *)
Join[{1,2},Range[2,200,2]] (* Harvey P. Dale, Jan 25 2019 *)
The graph
1---2---3
is a Tom graph: Tom can follow the sequence 2, 2 to guarantee that he catches Jerry.
The graph
1
/ \
2---3
is not a Tom graph: Jerry always has 2 vertices to go to, and whatever vertex Tom picks, Jerry can choose another to evade Tom.
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