A290884 -id:A290884 - cp's OEIS frontend

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

Rémy Sigrist, Aug 13 2017

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.

			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.

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

Cf. A290538, A290884, A290885.

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Aug 13 2017

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

See A290886 for the square of the norm of the n-th term of S.

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

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A290885

Cf. A290537, A290884, A290886.

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

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)(1+i) + c(2)(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.

Original entry on oeis.org

0, 1, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 30, 31, 34, 35, 36, 37, 40, 41, 86, 87, 90, 91, 92, 93, 96, 97, 102, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 120, 121, 126, 127, 130, 131, 132, 133, 136, 137, 150, 151

Offset: 1

Eugen Ionascu, Nov 15 2022

nonn

			6 is in the sequence since 6 = T^2 + T^3 + T^4 + T^5 + T^8, where T=1+i.

Problem 12335, American Mathematical Monthly, Vol. 129, issue 7, August-September 2022.

Wikipedia, Complex-base system.

Cf. A290884.

cpol[a_, b_] :=
Module[{u, uu, v, vv, p, pp, q, L, x, k, W}, u = a; v = b; p = {};
  W = {-2 - I, 0, -1 + 2*I};
  While[(u + 1)^2 + v^2 > 1,
   If[Mod[u + v, 2] ==
     0, {uu = (u + v)/2; vv = (v - u)/2; p = Prepend[p, 0]};,
    {uu = (u - 1 + v)/2; vv = (v + 1 - u)/2; p = Prepend[p, 1]}
    ]; u = uu; v = vv;
   ]; w = u + v*I; q = MemberQ[W, w]; L = Length[p];
  If[q == True, pp[x_] := Sum[p[[k]]*x^(L - k), {k, 1, L}],
   pp[x_] := "No writing"] ; {w, pp[1 + I], pp[T]}]

is(n)= while (n, if (n==I, return (0), real(n)%2==imag(n)%2, n = n/(1+I), n = (n-1)/(1+I));); return (1); \\ Rémy Sigrist, Nov 16 2022

More terms from Charles R Greathouse IV, Nov 15 2022

cp's OEIS Frontend

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.

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)(1+i) + c(2)(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)*(1+i) + c(2)*(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)(1+i) + c(2)(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.