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

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

Author

Rok Cestnik, Feb 23 2020

Keywords

• nonn
• base

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Python

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)

Formula

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.