

A290536


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


4



0, 1, 2, 1, 3, 2, 2, 0, 4, 3, 3, 0, 3, 1, 1, 2, 5, 4, 4, 0, 4, 1, 1, 4, 4, 2, 2, 2, 1, 1, 4, 6, 5, 5, 0, 5, 1, 1, 6, 5, 2, 2, 4, 3, 3, 8, 5, 3, 3, 2, 1, 1, 6, 0, 4, 3, 3, 4, 7, 6, 6, 0, 6, 1, 1, 8, 6, 2, 2, 6, 5, 5, 12, 6, 3, 3, 4, 3
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OFFSET

1,3


COMMENTS

See A290537 for the imaginary part of the nth term of S.
See A290538 for the square of the norm of the nth term of S.
The representation of the first terms of S in the complex plane has nice fractal features (see also Links section).
The sequence S is a "complex" variant of A232559.
The sequence S is a permutation of the Gaussian integers (Z[i]):
 let u be the function defined over Z[i] by z > z+1,
 let v be the function defined over Z[i] by z > z*(1+i),
 for m, n, o, p and q >= 0,
let f(m,n,o,p,q) = u^m(v(u^n(v(u^o(v(v(u^p(v(v(u^q(0)))))))))))
(where w^k denotes the kth iterate of w),
 f(m,0,0,0,0) = m, and any nonnegative integer x can be represented in this way for some m >= 0,
 f(m,n,0,0,0) = m+n + n*i, and any Gaussian integer x+y*i with 0 <= x and 0 <= y <= x can be represented in this way for some m and n >= 0,
 f(m,n,o,0,0) = f(m,n,0,0,0) + 2*o*i, and any Gaussian integer x+y*i with 0 < x and 0 <= y can be represented in this way for some m, n and o >= 0,
 f(m,n,o,p,0) = f(m,n,o,0,0)  4*p, and any Gaussian integer x+y*i with 0 <= y can be represented in this way for some m, n, o and p >= 0,
 f(m,n,o,p,q) = f(m,n,o,p,0)  8*q*i, and any Gaussian integer x+y*i can be represented in this way for some m, n, o, p and q >= 0,
 in other words, any Gaussian integer can be reached from 0 after a finite number of steps chosen in { u, v }, QED.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Representation of the first 100000 terms of S in the complex plane
Rémy Sigrist, Colorized representation of the first 100000 terms of S in the complex plane
Rémy Sigrist, PARI program for A290536
Wikipedia, Gaussian integer


EXAMPLE

S(1) = 0 by definition; so a(1) = 0.
S(1)+1 = 1 has not yet occurred; so S(2) = 1 and a(2) = 1.
S(1)*(i+i) = 0 has already occurred.
S(2)+1 = 2 has not yet occurred; so S(3) = 2 and a(3) = 2.
S(2)*(1+i) = 1+i has not yet occured; so S(4) = 1+i and a(4) = 1.
S(3)+1 = 3 has not yet occurred; so S(5) = 3 and a(5) = 3.


PROG

(PARI) See Links section.


CROSSREFS

Cf. A232559, A290537, A290538.
Sequence in context: A266756 A035181 A035151 * A277855 A136662 A023595
Adjacent sequences: A290533 A290534 A290535 * A290537 A290538 A290539


KEYWORD

sign,look


AUTHOR

Rémy Sigrist, Aug 05 2017


STATUS

approved



