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A316657
For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the real part of f(n).
11
0, 1, 0, -1, 0, 2, 3, 2, 1, 2, -1, 0, -1, -2, -1, -2, -1, -2, -3, -2, 1, 2, 1, 0, 1, 3, 4, 3, 2, 3, 5, 6, 5, 4, 5, 2, 3, 2, 1, 2, 1, 2, 1, 0, 1, 4, 5, 4, 3, 4, -4, -3, -4, -5, -4, -2, -1, -2, -3, -2, -5, -4, -5, -6, -5, -6, -5, -6, -7, -6, -3, -2, -3, -4, -3
OFFSET
0,6
COMMENTS
See A316658 for the imaginary part of f.
See A316707 for the square of the modulus of f.
The function f has nice fractal features (see scatterplot in Links section).
It appears that f defines a bijection from the nonnegative integers to the Gaussian integers.
LINKS
Rémy Sigrist, Colored scatterplot of (a(n), A316658(n)) for n=0..5^8-1 (where the hue is function of n)
Wikipedia, Gaussian integer
FORMULA
a(5^n) = A139011(n) for any n >= 0.
a(3 * 5^n) = -A139011(n) for any n >= 0.
MATHEMATICA
a[n_] := Module[{d, z}, d = IntegerDigits[n, 5] // Reverse; z = Sum[ If[d[[i]]>0, (2+I)^(i-1)*I^(d[[i]]-1), 0], {i, 1, Length[d]}]; Re[z]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 06 2021, after PARI code *)
PROG
(PARI) a(n) = my (d=Vecrev(digits(n, 5)), z=sum(i=1, #d, if (d[i], (2+I)^(i-1) * I^(d[i]-1), 0))); real(z)
CROSSREFS
KEYWORD
sign,base
AUTHOR
Rémy Sigrist, Jul 09 2018
STATUS
approved