OFFSET
1,2
COMMENTS
From Andrew Howroyd, Mar 14 2023: (Start)
The following information was extracted from the Derive script and notes.
The values of x and y are not required to be integers or even real numbers.
Eliminating x and y from the equations gives:
a^3 - 3*a*b + 2*c = 0.
This is the Diophantine equation that is being solved. For a given b and c there will be three possibly complex roots. It is required to find those values of b and c where all three roots are integers.
The values for b are the integers of the form (d^2 + d*e + e^2)/3 where d and e are positive integers and d <= e. This sequence lists these values in order with repetition.
The corresponding values for c are d*e*(d+e)/2 given in A108940, and the three values for a are d, e and -(d+e).
(End)
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
James R FitzSimons, Derive program
PROG
(Derive) See Links section
(PARI) upto(blim)={my(L=List()); for(e=1, sqrtint(3*blim), for(d=1, e, my(b=(d^2+d*e+e^2)/3); if(b<=blim && !frac(b), listput(L, b)))); listsort(L); Vec(L)} \\ Andrew Howroyd, Mar 13 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
James R FitzSimons (cherry(AT)getnet.net), Jun 22 2007
EXTENSIONS
Terms a(42) and beyond from Andrew Howroyd, Mar 14 2023
STATUS
approved