A327976 Bitwise XOR of trajectories (centrally aligned) of rule 30, and its mirror image, rule 86, when both are started from a lone 1-bit, with the latter delayed by one step: a(n) = A110240(n) XOR 2*A265281(n-1).

5, 23, 73, 359, 1233, 6143, 19225, 93495, 325729, 1518895, 4833289, 23453735, 81443089, 398815039, 1271974489, 6168932215, 21231239841, 99197620591, 314863189193, 1541326542823, 5312985402193, 26258203294847, 82884499362201, 400683454289591, 1406328980294113, 6532877164215983, 20744329255918985, 100303645024039591

Offset: 1

Author

Antti Karttunen, Oct 04 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1024
• Antti Karttunen, Terms a(1)-a(256) drawn as binary strings, with 1 bit = 3x3 pixels resolution
• Antti Karttunen, Terms a(1)-a(1024) drawn as binary strings, with 1 bit = 1 pixel resolution
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A110240, A265281, A269160, A269161, A030101, A327974 (gives the middle bit), A328108 (binary weight).

Cf. also A327971, A327972, A327973, A328103, A328104 for other such combinations.

Programs

PARI

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A327973(n) = bitxor(A110240(n), 2*A110240(n-1));
A269161(n) = bitxor(4*n, bitor(2*n, n));
A265281(n) = if(!n,1,A269161(A265281(n-1)));
A327976(n) = bitxor(A110240(n), 2*A265281(n-1));
\\ Use this one for writing b-files:
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A327976write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b327976.txt", n, " ", bitxor(2*A030101(s), t)); s = t); };

Python

def A269160(n): return(n^((n<<1)|(n<<2)))
def A269161(n): return((n<<2)^((n<<1)|n))
def genA327976():
    '''Yield successive terms of A327976.'''
    s1 = 1
    s2 = 1
    while True:
       s1 = A269160(s1)
       yield (s1^(s2<<1))
       s2 = A269161(s2)

Formula

a(n) = A110240(n) XOR 2*A265281(n-1) = A110240(n) XOR 2*A030101(A110240(n-1)).