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.

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, 34, 25, 52, 41, 50, 41, 32, 25, 18, 13, 20, 13, 10, 5, 40, 29, 26, 17, 36, 25, 26, 17, 80, 65, 58, 45, 68, 53, 50, 37, 104, 85, 82, 65, 100, 81, 82, 65, 64, 65, 50, 53

Offset

1,3

Comments

See A290884 for the real part of the n-th term of S, and additional comments.

See A290885 for the imaginary part of the n-th term of S.

a(n) tends to infinity as n tends to infinity.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A290886

Formula

a(n) = A290884(n)^2 + A290885(n)^2.

Example

Let f be the function z -> z * (1+i), and g the function z -> (z-1) * (1+i) + 1.
S(1) = 0 by definition; so a(1) = 0.
f(S(1)) = 0 has already occurred.
g(S(1)) = -i has not yet occurred; so S(2) = -i and a(2) = 1.
f(S(2)) = 1 - i has not yet occurred; so S(3) = 1 - i and a(3) = 2.
g(S(2)) = 1 - 2*i has not yet occurred; so S(4) = 1 - 2*i and a(4) = 5.
f(S(3)) = 2 has not yet occurred; so S(5) = 2 and a(5) = 4.
g(S(3)) = 2 - i has not yet occurred; so S(6) = 2 - i and a(6) = 5.
f(S(4)) = 3 - i has not yet occurred; so S(7) = 3 - i and a(7) = 10.
g(S(4)) = 3 - 2*i has not yet occurred; so S(8) = 3 - 2*i and a(8) = 13.

Mathematica

Table[Abs[FromDigits[IntegerDigits[n, 2], 1 + I]]^2, {n, 0, 100}] (* IWABUCHI Yu(u)ki, Jan 01 2023 *)

Prog

(PARI) See Links section.
(PARI) a(n) = norm(subst(Pol(binary(n-1)), 'x, I+1)); \\ Kevin Ryde, Apr 08 2020

Crossrefs

Cf. A290538, A290884, A290885.

Sequence in context: A197288 A053424 A184617 * A163809 A075771 A198813

Adjacent sequences: A290883 A290884 A290885 * A290887 A290888 A290889

Keyword

nonn,look

Author

Rémy Sigrist, Aug 13 2017

Status

approved