

A306514


Decimal representation of binary numbers with string structure 10s00, s in {0,1}*, such that it results in a nonpalindromic cycle of length 4 in the Reverse and Add! procedure in base 2.


4



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

1,1


COMMENTS

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 contextfree 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((k8)/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)binarydigit number (see A306515). Each mbinarydigit 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.


LINKS

A.H.M. Smeets, Table of n, a(n) for n = 1..6976


FORMULA

a(n) = 0 (mod 12).


EXAMPLE

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).


CROSSREFS

Cf. A061561, A103897, A306515, A306516, A306517.
Sequence in context: A260705 A295596 A287118 * A044416 A044797 A072589
Adjacent sequences: A306511 A306512 A306513 * A306515 A306516 A306517


KEYWORD

nonn,base


AUTHOR

A.H.M. Smeets, Feb 21 2019


STATUS

approved



