  This paper further generalizes a recent result of Shannon and Ollerton who
resurrected an old identity due to Ledin. This paper generalizes the
Ledin-Shannon-Ollerton result to all the metallic sequences. The results give
closed formulas for the sum of products of powers of the first $n$ integers
with the first $n$ members of the metallic sequence. Three key innovations of
this paper are i) reducing the proof of the generalization to the solution of a
system of 4 simultaneous recursions; ii) use of the shift operation to prove
equality of polynomials; and iii) new OEIS sequences arising from the
coefficients of the four polynomial families satisfying the 4 simultaneous
recursions.
