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

Author

Rémy Sigrist, Aug 13 2017

Keywords

• nonn
• look

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.

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A290886

Crossrefs

Cf. A290538, A290884, A290885.

Programs

Mathematica

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

PARI

See Links section.

PARI

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

Formula

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