  There is a growing literature on sums of reciprocals of polynomial functions
of recurrence relations with constant coefficients and fixed depth, such as
Fibonacci and Tribonacci numbers, products of such numbers, and balancing
numbers (numbers $n$ such that the sum of the integers less than $n$ equals the
sum of the $r$ integers immediately after, for some $r$ which is called the
balancer of $n$; If $n$ is included in the summation, we have the cobalancing
numbers, and $r$ is called the cobalancer of $n$). We generalize previous work
to reciprocal sums of depth two recurrence sequences with arbitrary
coefficients and the Tribonacci numbers, and show our method provides an
alternative proof of some existing results.
  We define $(a,b)$ balancing and cobalancing numbers, where $a$ and $b$ are
constants that multiply the left-hand side and right-hand side respectively,
and derive recurrence relations describing these sequences. We show that for
balancing numbers, the coefficients $(3,1)$ is unique such that every integer
is a $(3,1)$ balancing number, and proved there does not exist an analogous set
of coefficients for cobalancing numbers. We also found patterns for certain
coefficients that have no balancing or cobalancing numbers.
