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 A339602 a(n) = (a(n-2) XOR A030101(a(n-1))) + 1, a(0) = 0, a(1) = 1. 1
 0, 1, 2, 1, 4, 1, 6, 3, 6, 1, 8, 1, 10, 5, 16, 5, 22, 9, 32, 9, 42, 29, 62, 3, 62, 29, 42, 9, 36, 1, 38, 25, 54, 3, 54, 25, 38, 1, 40, 5, 46, 25, 62, 7, 58, 17, 44, 29, 60, 19, 38, 11, 44, 7, 44, 11, 34, 27, 58, 13, 50, 31, 46, 3, 46, 31, 50, 13, 58, 27, 34 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Is this sequence periodic? The related sequence A114375 was found to be nonperiodic, however the same argument does not hold here as the bitreversal operation used here maps different values onto the same. E.g. 111000 -> 111 111 -> 111. Every time this sequence develops a not yet seen value a(n) = 2^m, the space of combinations increases from (2^(m-1))^2 to (2^m)^2 reducing the probability to hit an already seen pair of values for [a(n-2),a(n-1)]. This sequence contains some palindromic parts. Example a(55)...a(72): 11, 34, 27, 58, 13, 50, 31, 46, 3, 46, 31, 50, 13, 58, 27, 34, 11. a(n) <> a(n-1). a(2k) = 2m. a(2k+1) = 2m+1. a(n) = 2^k, k > 0 for each k will exist only once in this sequence, if it is never periodic. In this case the 2^k will be in increasing sequence ordered. Conjecture: Let p be an odd number, then a(n) = p will be more frequently found in this sequence than a(n) = p+1. (tested for n=0..10^7 with primes > 2, but seems to be true for all odd too). LINKS Robert Israel, Table of n, a(n) for n = 0..10000 Thomas Scheuerle, Interesting staircase pattern in this sequence. EXAMPLE a(5) = 1 binary: 1; a(6) = 6 binary: 110, binary bitreversed: 11; so a(7) = binary: (001 XOR 11)+1 = 11 decimal: 3. MAPLE bitrev:= proc(n) local L, i;   L:= convert(n, base, 2);   add(L[-i]*2^(i-1), i=1..nops(L)) end proc: A:= Array(0..100): A[0]:= 0: A[1]:= 1: for n from 2 to 100 do   A[n]:= Bits:-Xor(A[n-2], bitrev(A[n-1]))+1 od: seq(A[i], i=0..100); # Robert Israel, Dec 25 2020 MATHEMATICA f[n_] := FromDigits[Reverse @ IntegerDigits[n, 2], 2]; a[0] = 0; a[1] = 1; a[n_] := a[n] = BitXor[a[n - 2], f[a[n - 1]]] + 1; Array[a, 100, 0] (* Amiram Eldar, Dec 10 2020 *) PROG (MATLAB) function a = calc_A339602(length)     % a(0) = 0  not in output of program     a(1) = 1; % part of definition     an_2 = 0; % a(0)     an_1 = a(1);     for n = 2:length         an_1_old = an_1;         an_1 = bitxor(an_2, bitreverse(an_1))+1;         an_2 = an_1_old;         a(n) = an_1;     end end function r = bitreverse(k) % A030101(k)     r = 0;     m = floor(log2(k))+1;     for i = 1:m         r = bitset(r, m-i+1, bitget(k, i));     end end (PARI) f(n) = fromdigits(Vecrev(binary(n)), 2); \\ A030101 lista(nn) = {my(x=0, y=1); print1(x, ", ", y, ", "); for (n=2, nn, z = bitxor(x, f(y)) +1; print1(z, ", "); x = y; y = z; ); } \\ Michel Marcus, Dec 10 2020 CROSSREFS Cf. A030101, A114375. Sequence in context: A341857 A292403 A271773 * A277127 A332792 A118275 Adjacent sequences:  A339599 A339600 A339601 * A339603 A339604 A339605 KEYWORD nonn,base,look AUTHOR Thomas Scheuerle, Dec 09 2020 STATUS approved

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Last modified June 13 09:02 EDT 2021. Contains 344981 sequences. (Running on oeis4.)