A290536 -id:A290536 - cp's OEIS frontend

A290537 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 imaginary part of the n-th term of S.

0, 0, 0, 1, 0, 2, 1, 2, 0, 3, 2, 4, 1, 3, 2, 2, 0, 4, 3, 6, 2, 5, 4, 4, 1, 4, 3, 4, 3, 2, 0, 0, 5, 4, 8, 3, 7, 6, 6, 2, 6, 5, 6, 5, 4, 0, 1, 5, 4, 6, 5, 4, 2, 3, 2, 1, 0, -4, 0, 6, 5, 10, 4, 9, 8, 8, 3, 8, 7, 8, 7, 6, 0, 2, 7, 6, 8, 7, 6, 2, 5, 2, 1, 0, -8, 1

Offset: 1

Rémy Sigrist, Aug 05 2017

See A290536 for the real part of the n-th term of S and additional comments.

See A290538 for the square of the norm of the n-th term of S.

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

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A290537

Cf. A290536, A290538.

PARI
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\\ See Links section.
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A290538 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 square of the norm of the n-th term of S.

Original entry on oeis.org

0, 1, 4, 2, 9, 8, 5, 4, 16, 18, 13, 16, 10, 10, 5, 8, 25, 32, 25, 36, 20, 26, 17, 32, 17, 20, 13, 20, 10, 5, 16, 36, 50, 41, 64, 34, 50, 37, 72, 29, 40, 29, 52, 34, 25, 64, 26, 34, 25, 40, 26, 17, 40, 9, 20, 10, 9, 32, 49, 72, 61, 100, 52, 82, 65, 128, 45, 68

Offset: 1

Rémy Sigrist, Aug 05 2017

See A290536 for the real part of the n-th term of S and additional comments.
See A290537 for the imaginary part of the n-th term of S.
a(n) tends to infinity as n tends to infinity.

			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) = 4.
S(2)*(1+i) = 1+i has not yet occurred; so S(4) = 1+i and a(4) = 2.
S(3)+1 = 3 has not yet occurred; so S(5) = 3 and a(5) = 9.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A290538

Cf. A290536, A290537.

PARI
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\\ See Links section.
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a(n) = A290536(n)^2 + A290537(n)^2.

A290884 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 real part of the n-th term of S.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, -4, -4, -3, -3, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, -4, -4, -3, -3, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, -8, -8, -7, -7, -6, -6, -5, -5, -6, -6, -5

Offset: 1

Rémy Sigrist, Aug 13 2017

See A290885 for the imaginary part of the n-th term of S.
See A290886 for the square of the norm of the n-th term of S.
This sequence is a variant of A290536.
The representation of the first terms of S in the complex plane has nice fractal features, and looks like a Dragon curve (see also Links section).
The building of this sequence is close to that of the Twindragon (see Wikipedia link).
The sequence S' built with the same rules but with the initial term S'(1) = 1 seems to be the complement of S; the set of elements of S is the image of the set of elements of S' with respect to the symmetry z -> 1 - z.
From Rémy Sigrist, Jul 10 2018: (Start)
For any n >= 0 with binary expansion Sum_{k=0..h} b_k * 2^k, let g(n) = Sum_{k=0..h} b_k * (1+i)^k (where i denotes the imaginary unit).
Apparently, g(n) = i * a(n+1) - A290885(n+1) for any n >= 0.
The function g has similarities with the function f defined in A316657.
(End)

			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) = 0.
f(S(2)) = 1 - i has not yet occurred; so S(3) = 1 - i and a(3) = 1.
g(S(2)) = 1 - 2*i has not yet occurred; so S(4) = 1 - 2*i and a(4) = 1.
f(S(3)) = 2 has not yet occurred; so S(5) = 2 and a(5) = 2.
g(S(3)) = 2 - i has not yet occurred; so S(6) = 2 - i and a(6) = 2.
f(S(4)) = 3 - i has not yet occurred; so S(7) = 3 - i and a(7) = 3.
g(S(4)) = 3 - 2*i has not yet occurred; so S(8) = 3 - 2*i and a(8) = 3.

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, Colorized representation of the first 1000000 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 A290884
Wikipedia, Dragon curve
Wikipedia, Twindragon

Cf. A290536, A290885, A290886, A316657.

PARI
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See Links section.
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a(n) = imag(subst(Pol(binary(n-1)),'x,I+1)); \\ Kevin Ryde, Apr 04 2020

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A290537 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 imaginary part of the n-th term of S.

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A290538 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 square of the norm of the n-th term of S.

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Author

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Comments

Examples

Links

Crossrefs

Programs

PARI

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A290884 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 real part of the n-th term of S.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI