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A318722
Let f(0) = 0 and f(t*4^k + u) = i^t * ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the real part of g(n).
3
-1, -1, 0, -1, -2, -2, -2, -2, -1, 0, 1, 1, -2, -3, -3, -1, -1, -2, -3, -4, -4, -4, -4, -3, -3, -3, -2, -3, -4, -4, -4, -4, -3, -2, -1, -1, 1, 2, 2, 0, 0, 1, 2, 3, 3, 3, 3, 2, -4, -5, -5, -3, -3, -4, -5, -6, -6, -6, -6, -5, -2, -2, -3, -2, -1, -1, -1, -1, -2
OFFSET
0,5
COMMENTS
See A318723 for the imaginary part of g.
See A318724 for the square of the modulus of g.
This sequence can be computed by considering the base 4 representation of A042968, hence the keyword base.
The function g runs uniquely through the set of Gaussian integers z such that Re(z) < 0 or Im(z) < 0.
The function g is related to the numbering of the cells in a Chair tiling (see representation of g(n) in Links section).
This sequence has similarities with A316657.
LINKS
Rémy Sigrist, Colored scatterplot of (a(n), A318723(n)) for n = 0..3*4^9-1 (where the hue is function of n)
Rémy Sigrist, Colored scatterplot of (a(n), A318723(n)) for n = 0..3*4^9-1 (where the color is function of the sum of digits of A042968(n) in base 4)
Tilings Encyclopedia, Chair
FORMULA
a(n) = A318723(n) iff the base 4 representation of A042968(n) contains only 0's and 2's.
If A048647(A042968(m)) = A042968(n), then a(m) = A318723(n) and A318723(m) = a(n).
PROG
(PARI) a(n) = my (d=Vecrev(digits(1+n+n\3, 4)), z=0); for (k=1, #d, if (d[k], z = I^d[k] * (-z + (1+I) * 2^(k-1)))); real((z-1-I)/2)
CROSSREFS
KEYWORD
sign,base
AUTHOR
Rémy Sigrist, Sep 02 2018
STATUS
approved