cp's OEIS Frontend

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

%I A117998 #47 Feb 16 2025 08:33:00
%S A117998 1,6,20,120,272,1632,5440,32640,65792,394752,1315840,7895040,17895424,
%T A117998 107372544,357908480,2147450880,4295032832,25770196992,85900656640,
%U A117998 515403939840,1168248930304,7009493581824,23364978606080
%N A117998 Decimal number generated by the binary bits of the n-th generation of the Rule 102 elementary cellular automaton.
%C A117998 Central diagonal of A099884 when viewed as a square array. Thus also a subsequence of A118666. - _Antti Karttunen_, Jan 18 2020
%H A117998 Robert Price, <a href="/A117998/b117998.txt">Table of n, a(n) for n = 0..999</a>
%H A117998 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule102.html">Rule 102</a>
%H A117998 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A117998 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A117998 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>
%F A117998 It appears that a(n) = A099885(2*n). - _Peter Bala_, Feb 01 2017
%F A117998 From _Antti Karttunen_, Jan 19 2020: (Start)
%F A117998 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.
%F A117998 a(n) = A000079(n) * A001317(n). [See Eric Weisstein's World of Mathematics -link]
%F A117998 a(0) = 1; for n > 0, a(n) = A048726(a(n-1)).
%F A117998 (End)
%e A117998 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; ...
%t A117998 NestList[BitXor[4#,2#]&,1,50] (* _Paolo Xausa_, Oct 04 2023 *)
%o A117998 (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.
%o A117998 (Python)
%o A117998 def A117998(n): return sum((bool(~n&n-k)^1)<<k for k in range(n+1))<<n # _Chai Wah Wu_, May 03 2023
%Y A117998 Iterates of A048726, starting from a(0) = 1.
%Y A117998 Central diagonal of A099884. Bisection of A099885. Subsequence of A118666.
%Y A117998 Cf. A000079, A001317, A193231.
%K A117998 nonn,base
%O A117998 0,2
%A A117998 _Eric W. Weisstein_, Apr 08 2006