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A306516 a(n) is the decimal representation of the binary number 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 and is not in the trajectory of any m < a(n).

%I A306516 #34 Aug 15 2025 09:11:41
%S A306516 84,324,2628,10836,21060,21684,23556,43188,43668,86388,86964,166212,
%T A306516 168516,172404,173364,332484,344820,345972,348564,664260,666180,
%U A306516 688884,690804,695220,1328580,1330884,1348164,1362564,1377780,1380084,1385268,1387092,1510020,2655684,2659524,2726532,2754036
%N A306516 a(n) is the decimal representation of the binary number 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 and is not in the trajectory of any m < a(n).
%C A306516 It is conjectured that all terms in this sequence are derived from S -> S_a -> 10 T_a 00 -> 10 T_a0 00 (in decimal representation here) when we represent the cyclic patterns by the context-free grammar with production rules:
%C A306516   S -> S_a | S_b | S_c | S_d
%C A306516   S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0;
%C A306516   S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0;
%C A306516   S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0;
%C A306516   S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0;
%C A306516   T_a0, T_b0, T_c0 and T_d0 being some terminating strings.
%C A306516 It is observed that numbers of the form 81*2^k + 2^floor((k+6)/2) + 2^6*(2^floor((k-8)/2) - 1) + 4 for k >= 11 (i.e. numbers lower(k) as defined in A306514) are all included in this sequence. By assuming this conjecture is true, it is easily to see that the number of seeds for the Reverse and Add! are infinite in base 2.
%C A306516 From _A.H.M. Smeets_, May 30 2019: (Start)
%C A306516 Several subsets of this sequence can be defined, each of them proving that there exists an infinite number of seeds in the Reverse and Add! procedure in base 2:
%C A306516 In the string representations, + stands for concatenation and ^ stands for repeated concatenation.
%C A306516 Example 1:
%C A306516 With f_1(n) = 81*2^n + 12*2^floor(n/2) - 60, {f_1(n) | n >= 11} is a proper subset.
%C A306516 String representation for the binary representation of f_1(n): 1010001 + (0)^floor((n-11)/2) + 0010 + (1)^floor((n-10)/2) + 1000100 for n >= 11.
%C A306516 f_1(n) = 3*f_1(n-1) - 6*f_1(n-3) + 4*f_1(n-4) for n > 3, f_1(0) = 33, f_1(1) = 114, f_1(2) = 288, f_1(3) = 612 (from _Colin Barker_).
%C A306516 f_1(n) = (-60 + 81*2^n + 3*2^((1+n)/2)*(1+(-1)^(n+1) + sqrt(2) + (-1)^n*sqrt(2))) (from _Colin Barker_).
%C A306516 G.f. for f_1: 3*(11 + 5*x - 18*x^2 - 18*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)) (from _Colin Barker_).
%C A306516 Example 2:
%C A306516 With f_2(n) = 32*8^n + 64*(8^n - 8)/7 + 68, {f_2(n) | n >= 2} is a proper subset.
%C A306516 String representation for the binary representation of f_2(n): 10 + (100)^(n-2) + 1000100 for n >= 2.
%C A306516 f_2(n) = 8*f_2(n-1) + 36 for n > 0, a(0) = 0.
%C A306516 f_2(n) = 36*(-1 + 8^(n+1))/7 (from _Colin Barker_).
%C A306516 f_2(n) = 9*f_2(n-1) - 8*f_2(n-2) for n > 1, a(0) = 0, a(1) = 36 (from _Colin Barker_).
%C A306516 G.f. for f_2: 36 / ((1 - x)*(1 - 8*x)) (from _Colin Barker_).
%C A306516 Example 3:
%C A306516 f_3(n) = 369*2^n - 24*2^floor(n/2) +132, {f_3(n) | n >= 12} is a proper subset.
%C A306516 String representation for the binary representation of f_3(n): 101110000 + (1)^ceiling((n-12)/2) + 101 + (0)^floor((n-12)/2) + 010000100 for n >= 12.
%C A306516 Example 4:
%C A306516 f_4(n) = 21*2^n + 6*2^floor(n/2) - 12, {f_4(n) | n >= 12} is a proper subset.
%C A306516 String representation for the binary representation of f_4(n): 10101 + (0)^floor((n-9)/2) + 0010 + (1)^ceiling((n-9)/2) + 10100 for n >= 12.
%C A306516 f_4(n) = 2*f_4(n-1) + 12*(-1)^n for n >= 4. (End)
%H A306516 A.H.M. Smeets, <a href="/A306516/b306516.txt">Table of n, a(n) for n = 1..1701</a>
%K A306516 nonn,base
%O A306516 1,1
%A A306516 _A.H.M. Smeets_, Feb 21 2019