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Search: id:A124869
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| A124869 |
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Numerator of real part of (2*omega)^(-n) where omega = (-1 + i*sqrt(3))/2. |
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+0
4
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| 1, -1, -2, 13, 7, -79, 11, 307, -527, -481, 779, -3827, -11753, 42641, 4031, -245453
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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.
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REFERENCES
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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.
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LINKS
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Wikipedia, Quater-imaginary base.
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FORMULA
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a(n) = numerator(Re((-1 + i*sqrt(3))^n). (-1 + i*sqrt(3))^n = A124869(n)/A124870(n) + i*A124871(n)/A124872(n).
G.f.:(10+x)/(10+2*x+x^2) = 1-1/10*x-2/25*x^2+13/500*x^3+7/2500*x^4-79/25000*x^5+... . - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 08 2007
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EXAMPLE
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a(0) = 1 = numerator of Re((-1+3*i)^0) = 1/1 + 0*i.
a(1) = -1 = numerator of Re(1/(-1+3*i)) = -1/10 - i*3/10.
a(2) = -2 = numerator of Re((-1+3*i)^(-2)) = -2/25 + i*3/50.
a(3) = 13 = numerator of Re((-1+3*i)^(-3)) = 13/500 + i*9/500.
a(4) = 7 = numerator of Re((-1+3*i)^(-4)) = 7/2500 - i*6/625.
a(5) = -79 = numerator of Re((-1+3*i)^(-5)) = -79/25000 + i*3/25000.
a(6) = 11 = numerator of Re((-1+3*i)^(-6)) = 11/31250 + i*117/125000.
a(7) = 307 = numerator of Re((-1+3*i)^(-7)) = 307/1250000 - i*249/1250000.
a(8) = -527 = numerator of Re((-1+3*i)^(-8)) = -527/6250000 - i*21/390625.
a(9) = -481 = numerator of Re((-1+3*i)^(-9)) = -481/62500000 + i*1917/62500000.
a(10) = 779 = numerator of Re((-1+3*i)^(-10)) = 779/78125000 - i*237/312500000.
a(11) = -3827 = numerator of Re((-1+3*i)^(-11)) = -3827/3125000000 - i*9111/3125000000.
a(12) = -11753 = numerator of Re((-1+3*i)^(-12)) = -11753/15625000000 + i*1287/1953125000.
a(13) = 42641 = numerator of Re((-1+3*i)^(-13)) = 42641/156250000000 + i*24963/156250000000.
a(14) = 4031 = numerator of Re((-1+3*i)^(-14)) = 4031/195312500000 - i*76443/781250000000.
a(15) = -245453 = numerator of Re((-1+3*i)^(-15)) = -245453/7812500000000 + i* 28071/7812500000000.
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CROSSREFS
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Cf. A003627, A014612, A112770, A121307.
Adjacent sequences: A124866 A124867 A124868 this_sequence A124870 A124871 A124872
Sequence in context: A129733 A084160 A128155 this_sequence A065584 A075032 A032932
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KEYWORD
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easy,frac,more,sign
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AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 11 2006
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