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 A253062 Largest order of a rooted tree that does not contain a rooted caterpillar subtree of order n. 1

%I

%S 0,1,2,3,5,7,11,16,23,34,49,70,103,148,211,310,445,634,931,1336,1903,

%T 2794,4009,5710,8383,12028,17131,25150,36085,51394,75451,108256,

%U 154183,226354,324769,462550,679063,974308,1387651,2037190,2922925,4162954,6111571

%N Largest order of a rooted tree that does not contain a rooted caterpillar subtree of order n.

%H Stephan Brandt, Janina MÃ¼ttel, Dieter Rautenbach, <a href="http://dx.doi.org/10.1007/s00493-014-2899-4">The circumference of the square of a connected graph</a>, Combinatorica 34 (2014), no. 5, 547--559. MR3276436.

%F See Maple code.

%F Conjectures from _Colin Barker_, Feb 21 2015: (Start)

%F a(n) = a(n-1)+3*a(n-3)-3*a(n-4) for n>10.

%F G.f.: -x^2*(x^8-x^7+x^6-x^5+x^4+x^3-x^2-x-1) / ((x-1)*(3*x^3-1)).

%F (End)

%p f:=proc(k,i)

%p if i=1 then (23*3^k-1)/2

%p elif i=2 then (33*3^k-1)/2

%p else (47*3^k-1)/2; fi;

%p end;

%p g:=proc(n) local r,s;

%p s := (n mod 3); if s=0 then s:=s+3; fi; r:=(n-s)/3;

%p f(r-2,s);

%p end;

%p a:=[0,1,2,3,5,7,11,16,23];

%p for n from 10 to 50 do a:=[op(a),g(n)]; od;

%Y Cf. A253063.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Jan 21 2015

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Last modified April 5 19:11 EDT 2020. Contains 333257 sequences. (Running on oeis4.)