cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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 n-th term of S.

Original entry on oeis.org

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
Offset: 1

Views

Author

Rémy Sigrist, Aug 05 2017

Keywords

Comments

See A290537 for the imaginary part of the n-th term of S.
See A290538 for the square of the norm of the n-th 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 k-th 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.

Examples

			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 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) = 3.

Links

Crossrefs

Cf. A232559, A290537, A290538.

Programs

  • PARI
    See Links section.

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.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Aug 05 2017

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A290536, A290538.

Programs

  • PARI
    \\ See Links section.

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.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Aug 13 2017

Keywords

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

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.
Showing 1-3 of 3 results.

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