0,2

Arises in computing a cube root of unity (Eisenstein integer) analogue of the quater-imaginary numeral system of D. E. Knuth (1955), which is a non-standard positional numeral system which uses the imaginary number 2i as base. By analogy with the quaternary numeral system, the quater-imaginary system can represent every complex number using only the digits 0, 1, 2, and 3, without a unary negative or positive sign. Eisenstein integers are of the form a + b*omega, where a and b are ordinary integers, and omega = (-1 + i*sqrt(3))/2 is a cube root of 1, the other cube roots of 1 being 1 and omega^2 = (-1 - i*sqrt(3))/2. The analogue of Fermat's 4n+1 Theorem for Eisenstein integers is that a prime p can be written in the form a^2 - a*b + b^2 = (a + b*omega)*(a + b*omega^2) iff 3 does not divide (p+1). These are precisely the primes of the form 3*m^2 + n^2 (Cuban primes and their negatives); see Conway. Eisenstein integers are complex numbers that are also members of the imaginary quadratic field Q(sqrt -3) = Z[omega]. The sums, differences, and products of Eisenstein integers is another Eisenstein integer.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.

D. E. Knuth. The Art of Computer Programming. Volume 2, 3rd Edition. Addison-Wesley. pp. 205, "Positional Number Systems" [first proposed by Knuth in 1955, submitted to a high-school science talent search].

Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

Wikipedia, Quater-imaginary base.

a(n) = numerator(Im((-1 + i*sqrt(3))^n). (-1 + i*sqrt(3))^n = A124869(n)/A124870(n) + i*A124871(n)/A124872(n).

a(0) = 0 = numerator of Im((-1+3*i)^0) = 1/1 + 0*i.

a(1) = -3 = numerator of Im(1/(-1+3*i)) = -1/10 - i*3/10.

a(2) = 3 = numerator of Im((-1+3*i)^(-2)) = -2/25 + i*3/50.

a(3) = 9 = numerator of Im((-1+3*i)^(-3)) = 13/500 + i*9/500.

a(4) = -6 = numerator of Im((-1+3*i)^(-4)) = 7/2500 - i*6/625.

a(5) = 3 = numerator of Im((-1+3*i)^(-5)) = -79/25000 + i*3/25000.

a(6) = 117 = numerator of Im((-1+3*i)^(-6)) = 11/31250 + i*117/125000.

a(7) = -249 = numerator of Im((-1+3*i)^(-7)) = 307/1250000 - i*249/1250000.

a(8) = -21 = numerator of Im((-1+3*i)^(-8)) = -527/6250000 - i*21/390625.

a(9) = 1917 = numerator of Im((-1+3*i)^(-9)) = -481/62500000 + i*1917/62500000.

a(10) = -237 = numerator of Im((-1+3*i)^(-10)) = 779/78125000 - i*237/312500000.

a(11) = -9111 = numerator of Im((-1+3*i)^(-11)) = -3827/3125000000 - i*9111/3125000000.

a(12) = 1287 = numerator of Im((-1+3*i)^(-12)) = -11753/15625000000 + i*1287/1953125000.

a(13) = 24963 = numerator of Im((-1+3*i)^(-13)) = 42641/156250000000 + i*24963/156250000000.

a(14) = -76443 = numerator of Im((-1+3*i)^(-14)) = 4031/195312500000 - i*76443/781250000000.

a(15) = 28071 = numerator of Im((-1+3*i)^(-15)) = -245453/7812500000000 + i*28071/7812500000000.

Cf. A003627, A014612, A112770, A121307, A124869-A124872.

Adjacent sequences: A124868 A124869 A124870 this_sequence A124872 A124873 A124874

Sequence in context: A066572 A104195 A062131 this_sequence A065483 A019745 A064235

easy,frac,more,sign

Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 11 2006