Shadow Sequences of Integers: From Fibonacci to Markov and Back.

Moreover, for some "mysterious" reasons, the new sequence HT <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo stretchy="false">(</mo><msub><mi> </mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo> </mo><mi mathvariant="double-struck">N</mi></mrow></msub></math> ht turns out to be an integer sequence! Indeed, it follows directly from (4) that (5) can be rewritten without division as HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>a</mi><mo>'</mo></msup><mo>=</mo><mn>3</mn><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi></mrow></math> ht . If HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mi> </mi></math> ht is a square root of -1, then I A i is a complex number, called a Gaussian integer. But if HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mi> </mi></math> ht satisfies the condition HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi> </mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math> ht then I A i is called a dual number. [Extracted from the article]