A114375 -id:A114375 - cp's OEIS frontend

A332782 a(n) = (a(n-1) XOR a(n-5)) + 1, a(0) = a(1) = a(2) = a(3) = a(4) = 0.

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 8, 12, 9, 13, 9, 2, 15, 7, 11, 3, 2, 14, 10, 2, 2, 1, 16, 27, 26, 25, 25, 10, 18, 9, 17, 9, 4, 23, 31, 15, 7, 4, 20, 12, 4, 4, 1, 22, 27, 32, 37, 37, 52, 48, 17, 53, 17, 38, 23, 7, 51, 35, 6, 18, 22, 38, 6, 1, 20, 3, 38, 33, 33, 54, 54, 17, 49

Offset: 0

Rok Cestnik, Feb 23 2020

			a(5) = (a(4) XOR a(0)) + 1 = (0 XOR 0) + 1 = 0 + 1 = 1.
a(6) = (a(5) XOR a(1)) + 1 = (1 XOR 0) + 1 = 1 + 1 = 2.
a(10) = (a(9) XOR a(5)) + 1 = (101_2 XOR 001_2) + 1 = 100_2 + 1 = 101_2 = 5_10.

Alois P. Heinz, Table of n, a(n) for n = 0..65536.
G. Marsaglia, Xorshift RNGs, Journal of Statistical Software, Vol. 8, Issue 14, Jul 2003.
Wikipedia, Bitwise operation (XOR).
Wikipedia, Linear-feedback shift register.
Wikipedia, Xorshift.

Cf. A114375 (shift 1), A332780 (shift 2), A332781 (shift 3).

a:= proc(n) option remember; `if`(n<5, 0,
       Bits[Xor](a(n-1), a(n-5))+1)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 09 2020

Nest[Append[#, 1 + BitXor @@ #[[{-1, -5}]] ] &, ConstantArray[0, 5], 75] (* Michael De Vlieger, Feb 23 2020 *)

feedback_delay = 4
a = [0 for i in range(feedback_delay+1)]
for i in range(feedback_delay,100):
    a.append((a[i]^a[i-feedback_delay])+1)

A182536 a(0)=0, a(1)=1, a(n)=(a(n-1) XOR a(n-2)) + n.

Original entry on oeis.org

0, 1, 3, 5, 10, 20, 36, 55, 27, 53, 56, 24, 44, 65, 123, 73, 66, 28, 112, 127, 35, 113, 104, 48, 112, 89, 67, 53, 146, 196, 116, 207, 219, 53, 272, 328, 124, 345, 331, 57, 410, 460, 128, 375, 547, 897, 464, 640, 896, 305, 739, 1029, 1818, 852, 1156, 2055, 3259, 1269, 2184, 3256, 1132, 2321, 3515, 1257, 2450, 3516, 1136, 2575

Offset: 0

Alex Ratushnyak, May 04 2012

Cf. A114375, A182509.

prpr, prev = 0,1
for n in range(2,99):
    current = (prpr ^ prev) + n
    print(prpr, end=', ')
    prpr, prev = prev, current

a(0)=0, a(1)=1, a(n)=(a(n-1) XOR a(n-2)) + n, where XOR is the bitwise exclusive-or operator.

A332780 a(n) = (a(n-1) XOR a(n-3)) + 1, a(0) = a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 3, 2, 2, 2, 1, 4, 7, 7, 4, 4, 4, 1, 6, 3, 3, 6, 6, 6, 1, 8, 15, 15, 8, 8, 8, 1, 10, 3, 3, 10, 10, 10, 1, 12, 7, 7, 12, 12, 12, 1, 14, 3, 3, 14, 14, 14, 1, 16, 31, 31, 16, 16, 16, 1, 18, 3, 3, 18, 18, 18, 1, 20, 7, 7, 20, 20, 20, 1, 22, 3, 3, 22, 22, 22

Offset: 0

Rok Cestnik, Feb 23 2020

			a(3) = (a(2) XOR a(0)) + 1 = (0 XOR 0) + 1 = 0 + 1 = 1.
a(4) = (a(3) XOR a(1)) + 1 = (1 XOR 0) + 1 = 1 + 1 = 2.
a(6) = (a(5) XOR a(3)) + 1 = (11_2 XOR 01_2) + 1 = 10_2 + 1 = 11_2 = 3_10.

G. Marsaglia, Xorshift RNGs, Journal of Statistical Software, Vol. 8, Issue 14, Jul 2003.
Wikipedia, Bitwise operation (XOR).
Wikipedia, Linear-feedback shift register.
Wikipedia, Xorshift.

Cf. A114375 (shift 1), A332781 (shift 3), A332782 (shift 4).

Nest[Append[#, 1 + BitXor @@ #[[{-1, -3}]] ] &, ConstantArray[0, 3], 75] (* Michael De Vlieger, Feb 23 2020 *)

feedback_delay = 2
a = [0 for i in range(feedback_delay+1)]
for i in range(feedback_delay,100):
    a.append((a[i]^a[i-feedback_delay])+1)

a(3+7*i) = 1; i >= 0.
a(3+7*i-{1,2,3,6}) = 2*i; i >= 0.
a(3+7*2^j*(1+2*i)-{4,5}) = 2^(j+2)-1; j >= 0, i >= 0.

A332781 a(n) = (a(n-1) XOR a(n-4)) + 1, a(0) = a(1) = a(2) = a(3) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 6, 5, 7, 4, 3, 7, 1, 6, 6, 2, 4, 3, 6, 5, 2, 2, 5, 1, 4, 7, 3, 3, 8, 16, 20, 24, 17, 2, 23, 16, 2, 1, 23, 8, 11, 11, 29, 22, 30, 22, 12, 27, 6, 17, 30, 6, 1, 17, 16, 23, 23, 7, 24, 16, 8, 16, 9, 26, 19, 4, 14, 21, 7, 4, 11, 31, 25, 30, 22, 10, 20

Offset: 0

Rok Cestnik, Feb 23 2020

Exclusive or (XOR) is a natural operation for computers. When combined with delayed feedback it can yield very complex behavior, which can be utilized for pseudorandom number generation [Marsaglia 2003]. Such "linear-feedback shift register" schemes can be implemented with basic computer operations and are hence very fast. Their behavior is typically periodic with long periods. Analogous procedures are used to generate this sequence, which takes a delayed feedback XOR and adds 1 at each iteration. It appears to be growing, although existence of loops has not been excluded.

			a(4) = (a(3) XOR a(0)) + 1 = (0 XOR 0) + 1 = 0 + 1 = 1.
a(5) = (a(4) XOR a(1)) + 1 = (1 XOR 0) + 1 = 1 + 1 = 2.
a(8) = (a(7) XOR a(4)) + 1 = (100_2 XOR 001_2) + 1 = 101_2 + 1 = 110_2 = 6_10.

G. Marsaglia, Xorshift RNGs, Journal of Statistical Software, Vol. 8, Issue 14, Jul 2003.
Wikipedia, Bitwise operation (XOR).
Wikipedia, Linear-feedback shift register.
Wikipedia, Xorshift.

Cf. A114375 (shift 1), A332780 (shift 2), A332782 (shift 4).

Nest[Append[#, 1 + BitXor @@ #[[{-1, -4}]] ] &, ConstantArray[0, 4], 75] (* Michael De Vlieger, Feb 23 2020 *)

feedback_delay = 3
a = [0 for i in range(feedback_delay+1)]
for i in range(feedback_delay,100):
    a.append((a[i]^a[i-feedback_delay])+1)

A339602 a(n) = (a(n-2) XOR A030101(a(n-1))) + 1, a(0) = 0, a(1) = 1.

Original entry on oeis.org

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

Offset: 0

Thomas Scheuerle, Dec 09 2020

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).

			a(5) = 1 binary: 1; a(6) = 6 binary: 110, binary bitreversed: 11;
so a(7) = binary: (001 XOR 11)+1 = 11 decimal: 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Thomas Scheuerle, Interesting staircase pattern in this sequence.

Cf. A030101, A114375.

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

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

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 *)

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

cp's OEIS Frontend

A332782 a(n) = (a(n-1) XOR a(n-5)) + 1, a(0) = a(1) = a(2) = a(3) = a(4) = 0.

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A182536 a(0)=0, a(1)=1, a(n)=(a(n-1) XOR a(n-2)) + n.

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A332780 a(n) = (a(n-1) XOR a(n-3)) + 1, a(0) = a(1) = a(2) = 0.

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A332781 a(n) = (a(n-1) XOR a(n-4)) + 1, a(0) = a(1) = a(2) = a(3) = 0.

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A339602 a(n) = (a(n-2) XOR A030101(a(n-1))) + 1, a(0) = 0, a(1) = 1.

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