A117998 Decimal number generated by the binary bits of the n-th generation of the Rule 102 elementary cellular automaton.

1, 6, 20, 120, 272, 1632, 5440, 32640, 65792, 394752, 1315840, 7895040, 17895424, 107372544, 357908480, 2147450880, 4295032832, 25770196992, 85900656640, 515403939840, 1168248930304, 7009493581824, 23364978606080

Offset

0,2

Comments

Central diagonal of A099884 when viewed as a square array. Thus also a subsequence of A118666. - Antti Karttunen, Jan 18 2020

Links

Robert Price, Table of n, a(n) for n = 0..999

Eric Weisstein's World of Mathematics, Rule 102

Index entries for sequences related to binary expansion of n

Index entries for sequences related to cellular automata

Index entries for sequences related to polynomials in ring GF(2)[X]

Formula

It appears that a(n) = A099885(2*n). - Peter Bala, Feb 01 2017

From Antti Karttunen, Jan 19 2020: (Start)

Bala's observation is correct, and follows from the formula given below and from the fact that this is the central diagonal of square array A099884.

a(n) = A000079(n) * A001317(n). [See Eric Weisstein's World of Mathematics -link]

a(0) = 1; for n > 0, a(n) = A048726(a(n-1)).

(End)

Example

1; 1, 1, 0; 1, 0, 1, 0, 0; 1, 1, 1, 1, 0, 0, 0; 1, 0, 0, 0, 1, 0, 0, 0, 0; ...

Mathematica

NestList[BitXor[4#, 2#]&, 1, 50] (* Paolo Xausa, Oct 04 2023 *)

Prog

(PARI) A117998(n) = (subst(lift(Mod(1+'x, 2)^n), 'x, 2)<<n); \\ Antti Karttunen, Jan 19 2020, after Gheorghe Coserea's code for A001317.
(Python)
def A117998(n): return sum((bool(~n&n-k)^1)<<k for k in range(n+1))<<n # Chai Wah Wu, May 03 2023

Crossrefs

Iterates of A048726, starting from a(0) = 1.

Central diagonal of A099884. Bisection of A099885. Subsequence of A118666.

Cf. A000079, A001317, A193231.

Sequence in context: A151493 A036755 A045470 * A267855 A099840 A223045

Adjacent sequences: A117995 A117996 A117997 * A117999 A118000 A118001

Keyword

nonn,base

Author

Eric W. Weisstein, Apr 08 2006

Status

approved