cp's OEIS Frontend

A290886 Let S be the sequence generated by these rules: 0 is in S, and if z is in S, then z * (1+i) and (z-1) * (1+i) + 1 are in S (where i denotes the imaginary unit), and duplicates are deleted as they occur; a(n) = the square of the norm of the n-th term of S.

%I A290886 #24 Jan 29 2024 03:45:42
%S A290886 0,1,2,5,4,5,10,13,8,5,10,9,20,17,26,25,16,9,10,5,20,13,18,13,40,29,
%T A290886 34,25,52,41,50,41,32,25,18,13,20,13,10,5,40,29,26,17,36,25,26,17,80,
%U A290886 65,58,45,68,53,50,37,104,85,82,65,100,81,82,65,64,65,50,53
%N A290886 Let S be the sequence generated by these rules: 0 is in S, and if z is in S, then z * (1+i) and (z-1) * (1+i) + 1 are in S (where i denotes the imaginary unit), and duplicates are deleted as they occur; a(n) = the square of the norm of the n-th term of S.
%C A290886 See A290884 for the real part of the n-th term of S, and additional comments.
%C A290886 See A290885 for the imaginary part of the n-th term of S.
%C A290886 a(n) tends to infinity as n tends to infinity.
%H A290886 Rémy Sigrist, <a href="/A290886/b290886.txt">Table of n, a(n) for n = 1..10000</a>
%H A290886 Rémy Sigrist, <a href="/A290886/a290886.gp.txt">PARI program for A290886</a>
%F A290886 a(n) = A290884(n)^2 + A290885(n)^2.
%e A290886 Let f be the function z -> z * (1+i), and g the function z -> (z-1) * (1+i) + 1.
%e A290886 S(1) = 0 by definition; so a(1) = 0.
%e A290886 f(S(1)) = 0 has already occurred.
%e A290886 g(S(1)) = -i has not yet occurred; so S(2) = -i and a(2) = 1.
%e A290886 f(S(2)) = 1 - i has not yet occurred; so S(3) = 1 - i and a(3) = 2.
%e A290886 g(S(2)) = 1 - 2*i has not yet occurred; so S(4) = 1 - 2*i and a(4) = 5.
%e A290886 f(S(3)) = 2 has not yet occurred; so S(5) = 2 and a(5) = 4.
%e A290886 g(S(3)) = 2 - i has not yet occurred; so S(6) = 2 - i and a(6) = 5.
%e A290886 f(S(4)) = 3 - i has not yet occurred; so S(7) = 3 - i and a(7) = 10.
%e A290886 g(S(4)) = 3 - 2*i has not yet occurred; so S(8) = 3 - 2*i and a(8) = 13.
%t A290886 Table[Abs[FromDigits[IntegerDigits[n, 2], 1 + I]]^2, {n, 0, 100}] (* _IWABUCHI Yu(u)ki_, Jan 01 2023 *)
%o A290886 (PARI) See Links section.
%o A290886 (PARI) a(n) = norm(subst(Pol(binary(n-1)),'x,I+1)); \\ _Kevin Ryde_, Apr 08 2020
%Y A290886 Cf. A290538, A290884, A290885.
%K A290886 nonn,look
%O A290886 1,3
%A A290886 _Rémy Sigrist_, Aug 13 2017