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A358432
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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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0
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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
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OFFSET
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1,3
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REFERENCES
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Problem 12335, American Mathematical Monthly, Vol. 129, issue 7, August-September 2022.
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LINKS
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EXAMPLE
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6 is in the sequence since 6 = T^2 + T^3 + T^4 + T^5 + T^8, where T=1+i.
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MATHEMATICA
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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]}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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