OFFSET
1,2
COMMENTS
The b-file contains all solutions for 1<=a<=100 and 1<=x<=2000000, double-checked by APL and Maple programs.
Of the 224 solutions:
100 are of the form (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1));
100 are of the form (a, 2(2a+1), (2a+1)^2, (4a+1)(2a+1));
6 are of the form (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3)),
where F(n) is the n-th Fibonacci number and k is odd;
(1, 2, 1, 3) is of both forms (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1)) and (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3));
19 have the form (a, cs, c^2t, cu) but appear to be otherwise unrelated.
LINKS
Christopher Hunt Gribble, Table of n, a(n) for n = 1..896
EXAMPLE
The sequence is the result of concatenating the rows in the table of solutions for a, x, b and y. The table starts:
. a x b y
. 1 2 1 3
. 1 6 9 15
. 2 5 6 16
. 2 6 9 21
. 2 10 25 45
. 2 57 495 609
. 2 637 21463 22737
. 3 10 25 55
. 3 14 49 91
. 4 14 49 105
. 4 18 81 153
. 5 13 40 105
. 5 18 81 171
. 5 22 121 231
. 5 574 27881 30751
. 6 22 121 253
. 6 26 169 325
For a=7 the three solution quadruplets are (7,26,169,351), (7,30,225,435) and (7,1340,120400,129780).
MAPLE
for a to 100 do
for x to 2000000 do
y := ceil(sqrt(a*x^3));
b := y^2-a*x^3;
if b > 0 and y = a*x+b then
printf("%10d %10d %10d %10d\n", a, x, b, y);
end if;
end do;
end do;
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Adam Kertesz and Christopher Hunt Gribble, Sep 25 2015
STATUS
approved