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A330780 Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v^2 times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * m(k) for n >= 0 is injective (where {m(k)} corresponds to knight's moves, see Comments for precise definition). 4

%I

%S 1,2,2,2,2,3,3,3,3,3,3,3,4,3,3,4,4,4,4,4,4,4,5,4,4,4,4,4,4,5,4,4,5,5,

%T 5,5,5,5,5,6,5,5,5,5,6,5,5,5,5,5,5,6,5,5,5,5,5,6,5,6,6,6,6,6,6,6,7,6,

%U 6,6,6,6,8,6,6,6,6,6,6,8,6,6,6,6,6,6,7

%N Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v^2 times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * m(k) for n >= 0 is injective (where {m(k)} corresponds to knight's moves, see Comments for precise definition).

%C The sequence {m(k)} is 8-periodic:

%C m(1) = 2 + i,

%C m(2) = 1 + 2*i, m(3) | m(2)

%C m(3) = -1 + 2*i, * | *

%C m(4) = -2 + i, m(4) * | * m(1)

%C m(5) = -2 - i, ------+------

%C m(6) = -1 - 2*i, m(5) * | * m(8)

%C m(7) = 1 - 2*i, * | *

%C m(8) = 2 - i. m(6) | m(7)

%H Rémy Sigrist, <a href="/A330780/b330780.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A330780/a330780_1.png">Illustration of first steps</a>

%H Rémy Sigrist, <a href="/A330780/a330780.png">Representation of f(n) for n = 0..1000000 in the complex plane</a> (where the color is function of n)

%H Rémy Sigrist, <a href="/A330780/a330780_2.png">Colored representation of the variant where the value v can appear up to v^3 times</a>

%H Rémy Sigrist, <a href="/A330780/a330780.gp.txt">PARI program for A330780</a>

%e The first terms, alongside the correspond value of f(n), are:

%e n a(n) f(n)

%e -- ---- -------

%e 0 N/A 0

%e 1 1 2+i

%e 2 2 4+5*i

%e 3 2 2+9*i

%e 4 2 -2+11*i

%e 5 2 -6+9*i

%e 6 3 -9+3*i

%e 7 3 -6-3*i

%e 8 3 -6*i

%e 9 3 6-3*i

%e 10 3 9+3*i

%e 11 3 6+9*i

%e 12 3 12*i

%e See also illustration in Links section.

%o (PARI) See Links section.

%Y See A331004 and A331005 for the real and imaginary parts of f, respectively.

%Y See A330779 for another variant.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Dec 31 2019

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Last modified August 11 05:05 EDT 2020. Contains 336422 sequences. (Running on oeis4.)