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[talk]

## Sequence of the Day for October 28

A026150
 (n + 1) /
A002605
 (n)
: Convergents to
 2√  3
.
 $\left\{ \frac{1}{1}, \frac{4}{2}, \frac{10}{6}, \frac{28}{16}, \frac{76}{44}, \frac{208}{120}, \frac{568}{328}, \frac{1552}{2448}, \cdots \right\} \,$
Obviously these fractions can be expressed in lower terms. But by leaving them as is, I wish to highlight that both the numerators
 a (n)
and the denominators
 b (n)
are obtained by recurrence relations of order
 2
(actually, the recurrences are the same, only the initial conditions differ):
$a(0) = 1, a(1) = 1; \,$
$a(n) = 2 (a(n - 1) + a(n - 2)),\quad n \ge 2. \,$
$b(0) = 0, b(1) = 1; \,$
$b(n) = 2(b(n - 1) + b(n - 2)),\quad n \ge 2. \,$
(The
 0
th term of the sequence would be
 1 0
:= ∞
.)

Note also the recurrence (involving only the previous numerator/denominator)

$a(1) = 1, b(1) = 1; \,$
$a(n) = a(n-1) + 3 b(n-1),\quad n \ge 2; \,$
$b(n) = a(n-1) + b(n-1),\quad n \ge 2. \,$