A348653 - OEIS

%I #12 Oct 29 2021 13:32:03

%S 0,0,1,2,1,1,1,0,-1,-2,-1,-1,-1,2,2,3,4,3,3,3,2,1,0,1,1,1,5,5,6,7,6,6,

%T 6,5,4,3,4,4,4,8,8,9,10,9,9,9,8,7,6,7,7,7,3,3,4,5,4,4,4,3,2,1,2,2,2,1,

%U 1,2,3,2,2,2,1,0,-1,0,0,0,-1,-1,0,1,0,0

%N For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the imaginary 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 A348652 for the real part.

%C The function f defines a bijection from the nonnegative integers to the Gaussian integers.

%C The following diagram depicts g(d) for d = 0..12:

%C                       |

%C                       |    +

%C                       |     3

%C                       |

%C             +    +    +    +

%C              6    5   |4    2

%C                       |

%C          --------+----+----+-------

%C                   7   |0    1

%C                       |

%C                  +    +    +    +

%C                   8   |10   11   12

%C                       |

%C                  +    |

%C                   9   |

%H Rémy Sigrist, <a href="/A348653/b348653.txt">Table of n, a(n) for n = 0..2197</a>

%H Stephen K. Lucas, <a href="https://www.researchgate.net/publication/355680849_Base_2_i_with_digit_set_0_1_i">Base 2 + i with digit set {0, +/-1, +/-i}</a>, ResearchGate (October 2021).

%H Rémy Sigrist, <a href="/A348652/a348652.png">Colored representation of f for n = 0..13^5-1 in the complex plane</a> (the hue is function of n)

%F abs(a(13^k)) = A188982(k).

%o (PARI) g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }

%o a(n) = imag(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))

%Y See A316658 for a similar sequence.

%Y Cf. A188982, A348652.

%K sign,base

%O 0,4

%A _Rémy Sigrist_, Oct 27 2021