Journal of Integer Sequences, Vol. 20 (2017), Article 17.9.7 |

Département de Mathématiques

UQAM

Case Postale 8888, Succ. Centre-ville

Montréal, Québec H3C 3P8

Canada

**Abstract:**

Reutenauer and Kassel introduced a family
*P*_{n}(*q*) of polynomials
defined in terms of divisors of *n* on overlapped intervals. The
evaluation of *P*_{n}(*q*)
at roots of unity of order 2, 3, 4, 6 form
well-known integer sequences related to the number of integer solutions
of the equations *x*^{2} + *y*^{2} = *n*,
*x*^{2} + 2*y*^{2} = *n*,
and *x*^{2} + *xy* + *y*^{2}
= *n*. Also, *P*_{n}(1) is the
sum of divisors of *n*.
In this paper we
define a new family *L*_{n}(*q*)
of polynomials defined in terms of
divisors of *n* on overlapped intervals,
slightly modifying the
definition of *P*_{n}(*q*).
The values of *L*_{n}(*q*)
at *q* = 1 and *q* = -1
are related to the sum of divisors of *n* and to the number of integer
solutions of the equations
*x*^{2} + *xy* + *y*^{2} = *n*
and *x*^{2} + 3 *y*^{2} = *n*.

(Concerned with sequences A002324 A096936.)

Received September 16 2017;
revised versions received October 13 2017; October 21 2017.
Published in *Journal of Integer Sequences*, October 29 2017.

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