A264200 Numerator of sum of numbers in set g(n) generated as in Comments.

0, 1, 5, 19, 69, 235, 789, 2603, 8533, 27819, 90453, 293547, 951637, 3082923, 9983317, 32320171, 104617301, 338602667, 1095849301, 3546458795, 11477013845, 37141260971, 120193373525, 388957383339, 1258699445589, 4073250794155, 13181344109909, 42655780874923

Offset: 0

Author

Clark Kimberling, Nov 09 2015

Keywords

• nonn
• easy

Comments

Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:

(1) if x is in g(n-1), then x + 1 is in g(n); and

(2) if x is in g(n-1) and x < 2, then x/2 is in g(n).

The sum of numbers in g(n) is a(n)/2^(n-1).

Examples

			g(0) = {0}, sum = 0.
g(1) = {1}, sum = 1.
g(2) = {1/2,2/1}, sum = 5/4.
g(3) = {1/4,3/2,3/1}, sum = 19/8.
		

Crossrefs

Cf. A054123, A054124, A264201.

Programs

Mathematica

z = 30; x = 1/2; g[0] = {0}; g[1] = {1};
g[n_] := g[n] = Union[1 + g[n - 1], (1/2) Select[g[n - 1], # < 2 &]]
Table[g[n], {n, 0, z}]; Table[Total[g[n]], {n, 0, z}]
Numerator[Table[Total[g[n]], {n, 0, z}] ]

Formula

Conjecture: a(n) = 3*a(n-1) + 4*a(n-2) - 8*a(n-3) - 8*a(n-4).