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A264201 Numerator of sum of numbers in set g(n) generated as in Comments.

%I A264201 #10 Jul 23 2025 15:39:20
%S A264201 0,1,7,46,265,1519,8560,47578,264076,1461439,8075011,44596708,
%T A264201 246189961,1358762089,7498499272,41378660380,228330571360,
%U A264201 1259923712821,6952163820391,38361311420962,211673092313329,1167984733037851,6444783128779528,35561432547881926
%N A264201 Numerator of sum of numbers in set g(n) generated as in Comments.
%C A264201 Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
%C A264201 (1)  if x is in g(n-1), then x + 1 is in g(n); and
%C A264201 (2)  if x is in g(n-1) and x < 3, then x/3 is in g(n).
%C A264201 The sum of numbers in g(n) is a(n)/3^(n-1).
%F A264201 Conjecture: a(n) = 4*a(n-1) + 9*a(n-2) + 18*a(n-3) - 81*a(n-4) - 162*a(n-5) - 243*a(n-6).
%e A264201 g(0) = {0}, sum = 0.
%e A264201 g(1) = {1}, sum = 1.
%e A264201 g(2) = {1/3,2/1}, sum = 7/3.
%e A264201 g(3) = {1/9,2/3,4/3,3/1}, sum = 46/9.
%t A264201 z = 5; x = 1/3;  g[0] = {0}; g[1] = {1};
%t A264201 g[n_] := g[n] = Union[1 + g[n - 1], (1/3) Select[g[n - 1], # < 3 &]]
%t A264201 Table[g[n], {n, 0, z}]
%t A264201 Table[Total[g[n]], {n, 0, z}]
%t A264201 u = Numerator[Table[Total[g[n]], {n, 0, z}] ]
%Y A264201 Cf. A264200.
%K A264201 nonn,easy
%O A264201 0,3
%A A264201 _Clark Kimberling_, Nov 09 2015