Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences
By Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger
.pdf
.ps
.tex
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Neil Sloane's website, and arxiv.org )
Written: Sept. 18, 2016
Dedicated to Richard K. Guy (b. 30 Sept., 1916) on his (100-ε)-th birthday
As far as I know, Richard Guy is the longest-living
mathematician of all time (in the sense of Paul Erdős, where you die when you stop doing math),
since at almost one hundred years of age, he keeps producing great new math.
This tribute, joint with Neil Sloane, is dedicated to him. Keep up the good work, Richard!
Maple package
-
Pisot.txt,
a Maple package to discover, prove, and disprove, linear recurrence relations satisfied by Pisot sequences,
first discovered and discussed by David Boyd, D.G. Cantor, and his student P. Galyean.
Sample Input and Output Files for the Maple package Pisot.txt
If you want to see linear recurrences, with COMPLETE proofs of Pisot sequenes of the form PISOT(a,b) (with parameter 1/2)
for 2<=a<20, and a+2<=b<40
the input file generates the
output file.
If you want to see PROVED linear recurrences, (w/o proofs, to save disk-space) of Pisot sequenes of the form PISOT(a,b) (with parameter 1/2)
for 2<=a<30, and a+2<=b<600
the input file generates the
output file.
If you want to see PROVED linear recurrences, (w/o proofs, to save disk-space) of Pisot sequenes of the form PISOT(a,b) (with parameter 1/2)
for 2<=a<40, and a+2<=b<1000
the input file generates the
output file.
If you want to see a detailed, fully rigorous proof that the Pisot Sequence E(4,7), alias
OEIS sequence A010901 does indeed satisfy (for ALL n, not just ≤ 50000) the recurrence
a(n) = 2a(n-1) - a(n-2) + a(n-3)
the input file generates the
output file.
If you want to see several examples of Generalized Pisot sequences that satisfy linear recurrences, along with their terse proofs
the input file generates the
output file.
If you want to see many infinite families of Pisot sequenes that satisfy linear recurrences
the input file generates the
output file.
If you want to see the same output, but in TeX
the input file generates the
output file.
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page