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%I #7 Nov 24 2015 01:42:37
%S 0,1,5,19,69,235,789,2603,8533,27819,90453,293547,951637,3082923,
%T 9983317,32320171,104617301,338602667,1095849301,3546458795,
%U 11477013845,37141260971,120193373525,388957383339,1258699445589,4073250794155,13181344109909,42655780874923
%N Numerator of sum of numbers in set g(n) generated as in Comments
%C Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
%C (1) if x is in g(n-1), then x + 1 is in g(n); and
%C (2) if x is in g(n-1) and x < 2, then x/2 is in g(n).
%C The sum of numbers in g(n) is a(n)/2^(n-1).
%F Conjecture: a(n) = 3*a(n-1) + 4*a(n-2) - 8*a(n-3) - 8*a(n-4).
%e g(0) = {0}, sum = 0.
%e g(1) = {1}, sum = 1.
%e g(2) = {1/2,2/1}, sum = 5/4.
%e g(3) = {1/4,3/2,3/1}, sum = 19/8.
%t z = 30; x = 1/2; g[0] = {0}; g[1] = {1};
%t g[n_] := g[n] = Union[1 + g[n - 1], (1/2) Select[g[n - 1], # < 2 &]]
%t Table[g[n], {n, 0, z}]; Table[Total[g[n]], {n, 0, z}]
%t Numerator[Table[Total[g[n]], {n, 0, z}] ]
%Y Cf. A054123, A054124, A264201.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Nov 09 2015