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.

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

Author

Eugen Ionascu, Nov 15 2022

Keywords

• nonn

Examples

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

References

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

Links

• Wikipedia, Complex-base system.

Crossrefs

Cf. A290884.

Programs

Mathematica

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]}]

PARI

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

Extensions

More terms from Charles R Greathouse IV, Nov 15 2022