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A348652
For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the real part of Sum_{k >= 0} g(d_k)*(3+2*i)^k where g(0) = 0, and g(1+u+3*v) = (1+u*i)*i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348653 for the imaginary part.
3
0, 1, 1, 1, 0, -1, -2, -1, -1, -1, 0, 1, 2, 3, 4, 4, 4, 3, 2, 1, 2, 2, 2, 3, 4, 5, 1, 2, 2, 2, 1, 0, -1, 0, 0, 0, 1, 2, 3, -1, 0, 0, 0, -1, -2, -3, -2, -2, -2, -1, 0, 1, -2, -1, -1, -1, -2, -3, -4, -3, -3, -3, -2, -1, 0, -5, -4, -4, -4, -5, -6, -7, -6, -6, -6
OFFSET
0,7
COMMENTS
The function f defines a bijection from the nonnegative integers to the Gaussian integers.
The following diagram depicts g(d) for d = 0..12:
|
| +
| 3
|
+ + + +
6 5 |4 2
|
--------+----+----+-------
7 |0 1
|
+ + + +
8 |10 11 12
|
+ |
9 |
LINKS
Stephen K. Lucas, Base 2 + i with digit set {0, +/-1, +/-i}, ResearchGate (October 2021).
FORMULA
a(13^k) = A121622(k).
PROG
(PARI) g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }
a(n) = real(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))
CROSSREFS
See A316657 for a similar sequence.
Sequence in context: A318133 A068029 A158208 * A117274 A221650 A140883
KEYWORD
sign,base
AUTHOR
Rémy Sigrist, Oct 27 2021
STATUS
approved