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A306514
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Decimal representation of binary numbers with string structure 10s00, s in {0,1}*, such that it results in a non-palindromic cycle of length 4 in the Reverse and Add! procedure in base 2.
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4
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84, 180, 324, 360, 684, 744, 1416, 1488, 2628, 2904, 3024, 5580, 5904, 6048, 10836, 11400, 11952, 12192, 21060, 21684, 23220, 23448, 23556, 24096, 24384, 43188, 43668, 44604, 44748, 46248, 46260, 47376, 48480, 48960, 86388, 86964, 91272, 92520, 92532, 93108, 95592, 96048, 96264, 97344, 97920, 166212
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OFFSET
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1,1
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COMMENTS
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If the decimal representation of the binary string 10s00 is in the sequence, so is 101s000.
For binary representation see A306515.
This sequence is a subset of A066059.
These regular patterns can be represented by the context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0,
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0,
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0,
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0,
where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
The decimal representation of all binary numbers derived by S -> S_a -> 10 T_a 00 -> 10 T_a0 00 are given in sequence A306516, its binary representation in A306517.
Observed: all values are in the ranges lower(k)..upper(k), where lower(k) = 81*2^k + 2^floor((k+6)/2) + 2^6*(2^floor((k-8)/2) - 1) + 4, which holds for k >= 11, and upper(k) = 3*2^floor((k+4)/2)*(2^floor((k+7)/2) - 1), which holds for k >= 0; the number of terms in each successive range increases by about a factor of 4/3. All terms between lower(k) and upper(k) are represented by a (k+7)-binary-digit number (see A306515). Each m-binary-digit number will have a successive number of m+1 binary digits in the next range. About 1/4 of each obtained number in this sequence has a new unique cyclic trajectory (see A306516 and A306517), i.e., a cyclic trajectory not joining a previous cyclic trajectory, which explains the growth factor of 4/3 for each successive range.
All terms A061561(4k+2) for k >= 0 are included in this sequence.
All values in A103897(k+3) for k >= 0 are included in this sequence.
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LINKS
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FORMULA
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a(n) = 0 (mod 12).
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EXAMPLE
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a(45) = 97920 = upper(10)
The following burst of terms is from a(46) = 166212 = lower(11) up to and including a(60) = 196224 = upper(12).
The burst of terms corresponding with k = 28 is from lower(28) = 21743468484 = a(5276) up to and including upper(28) = 25769607168 = a(6976).
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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