Automating John P. D'Angelo's method to study Complete Polynomial Sequences
By Shalosh B. Ekhad and Doron Zeilberger
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.tex
Written: Nov. 3, 2021.
In a recent intriguing article by complex geometer (and several complex variabler) John P. D'Angelo
there was a surprising application of elementary (but very deep!) number theory to complex geometry.
What D'Angelo needed was the largest integer not representable as sum of distinct values of triangular numbers
(with analogous questions about other polynomial sequences). He demonstrated, in terms of a few lucid examples, how
to determine these numbers, and then rigorously prove that they are indeed correct. In this short note
we describe a Maple package that implements these ideas enabling, at least in principle, but often also
in practice, to determine these numbers fully automatically. In fact, we show that
the very same ideas can be turned into an algorithm that inputs an arbitrary integergenerating polynomial,
and positive integers a and C, and outputs the largest integer not representable as a sum
of distinct values of that polynomial with argument ≥a in at least C different ways.
(The cases that D'Angelo needed were a=0 and a=1 and C=1).
Maple package
Input and Output files for JPDA.txt

If you want to see an article with the largest integer NOT representable as a sum of
distinct values of (j+1)*(j+2)/2, for j>=j0, for j0 from 0 to 6, each with a completely rigorous
proof, following John D'Angelo's method described in his nice
paper,
the input file yields
the output file

If you want to see an article, following D'Angelo's method in the abovementioned paper, that
tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)/2, for j>=j0, for j0 from 0 to 5, in at least C different ways
for C from 1 to 5,
the input file yields
the output file

If you want to see an article, following D'Angelo's method in the abovementioned paper, that
tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)*(j+3)/6, for j>=j0,
for j0 from 0 to 5
the input file yields
the output file

If you want to see an article, following D'Angelo's method in the abovementioned paper, that
tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)*(j+3)*(j+4)/24, for j>=j0,
for j0 from 0 to 5
the input file yields
the output file

If you want to see an article, following D'Angelo's method in the abovementioned paper, that
tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)*(j+3)*(j+4)*(j+5)/120, for j>=0 (it happens to be 120838)
and for j>=1 (it happens to be 291217)
the input file yields
the output file

If you want to see a more terse article, describing (without detailed proofs) the largest integer
not representable in at least C different ways as distinct sums of the polynomial p(j)=(j+1)*(j+2)/2 with j>=j0,
for C from 1 to 5 and j0 from 1 to 5,
the input file yields
the output file

If you want to see an example of D'Angelo's algorithm to produce representations
the input file yields
the output file

If you want to see an article, following D'Angelo's method in the abovementioned paper, that
tells you the largest integer NOT representable as a sum of distinct values of j^{2}, for j>=j0, for j0 from 0 to 5, in at least C different ways
for C from 1 to 5,
the input file yields
the output file

If you want to see an article, following D'Angelo's method in the abovementioned paper, that
tells you the largest integer NOT representable as a sum of distinct values of j^{3}, for j>=j0, for j0 from 0 to 5, in at least C different ways
for C from 1 to 5,
the input file yields
the output file

If you want to the first 11 terms of the sequence: The largest positive ineteger not representable as a sum
of distinct values of (j+1)(j+2)/2 with j ≥ i, for i=0,1,2,...,
the input file yields
the output file

If you want to the first 11 terms of the sequence: The largest positive ineteger not representable as a sum
of distinct values of (j+1)(j+2)(j+3)/6 with j ≥ i, for i=0,1,2,...,
the input file yields
the output file

If you want to the first 11 terms of the sequence: The largest positive ineteger not representable as a sum
of distinct values of (j+1)(j+2)(j+3)(j+4)/24 with j ≥ i, for i=0,1,2,...,
the input file yields
the output file
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