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%I #43 May 07 2019 18:36:25
%S 84,180,324,360,684,744,1416,1488,2628,2904,3024,5580,5904,6048,10836,
%T 11400,11952,12192,21060,21684,23220,23448,23556,24096,24384,43188,
%U 43668,44604,44748,46248,46260,47376,48480,48960,86388,86964,91272,92520,92532,93108,95592,96048,96264,97344,97920,166212
%N 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.
%C If the decimal representation of the binary string 10s00 is in the sequence, so is 101s000.
%C For binary representation see A306515.
%C This sequence is a subset of A066059.
%C These regular patterns can be represented by the context-free grammar with production rules:
%C S -> S_a | S_b | S_c | S_d
%C S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0,
%C S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0,
%C S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0,
%C S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0,
%C where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
%C Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
%C 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.
%C 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.
%C All terms A061561(4k+2) for k >= 0 are included in this sequence.
%C All values in A103897(k+3) for k >= 0 are included in this sequence.
%H A.H.M. Smeets, <a href="/A306514/b306514.txt">Table of n, a(n) for n = 1..6976</a>
%F a(n) = 0 (mod 12).
%e a(45) = 97920 = upper(10)
%e The following burst of terms is from a(46) = 166212 = lower(11) up to and including a(60) = 196224 = upper(12).
%e The burst of terms corresponding with k = 28 is from lower(28) = 21743468484 = a(5276) up to and including upper(28) = 25769607168 = a(6976).
%Y Cf. A061561, A103897, A306515, A306516, A306517.
%K nonn,base
%O 1,1
%A _A.H.M. Smeets_, Feb 21 2019
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