A306514 - OEIS

A306514

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.

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 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.

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.