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COMMENTS
Conjecture: a(n)/n -> (3+sqrt(5))/2.
EXAMPLE
As a word, A285501 = 00110011..., in which 0 is in positions 1,2,5,6,9,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 8] (* A285501 *) Flatten[Position[s, 0]] (* A285502 *) Flatten[Position[s, 1]] (* A285503 *)
0-limiting word of the morphism 0->11, 1-> 0011.
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0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0
COMMENTS
The morphism 0->11, 1-> 0011 has two limiting words. If the number of iterations is even, the 0-word evolves from 0 -> 11 -> 00110011 -> 111100110011111100110011 -> 0011001100110011111100110011111100110011...; if the number of iterations is odd, the 1-word evolves from 0 -> 11 -> 00110011 -> 111100110011111100110011, as in A285504.
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 8] (* A285501 *) Flatten[Position[s, 0]] (* A285502 *) Flatten[Position[s, 1]] (* A285503 *)
5, 6, 9, 10, 17, 18, 21, 22, 29, 30, 33, 34, 41, 42, 45, 46, 49, 50, 53, 54, 57, 58, 61, 62, 69, 70, 73, 74, 81, 82, 85, 86, 89, 90, 93, 94, 97, 98, 101, 102, 109, 110, 113, 114, 121, 122, 125, 126, 133, 134, 137, 138, 145, 146, 149, 150, 157, 158, 161, 162
COMMENTS
Conjecture: a(n)/n -> (3+sqrt(5))/2.
EXAMPLE
As a word, A285504 = 11110011001..., in which 0 is in positions 5,6,9,10,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 8] (* A285501 *) Flatten[Position[s, 0]] (* A285502 *) Flatten[Position[s, 1]] (* A285503 *)
1, 2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 16, 19, 20, 23, 24, 25, 26, 27, 28, 31, 32, 35, 36, 37, 38, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 65, 66, 67, 68, 71, 72, 75, 76, 77, 78, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 105
COMMENTS
Conjecture: a(n)/n -> (1+sqrt(5))/2.
EXAMPLE
As a word, A285504 = 11110011001..., in which 1 is in positions 1,2,3,4,7,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 8] (* A285501 *) Flatten[Position[s, 0]] (* A285502 *) Flatten[Position[s, 1]] (* A285503 *)
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