OFFSET
1,1
COMMENTS
The square of the third leg is a sum of consecutive cubes (or one cube). See A126200, A217843. In the Pythagorean triple {325,91,312}, 312^2 = 14^3 + 15^3 + ... + 25^3 = 97344.
It is possible for both of the legs to be triangular numbers as well as the hypotenuse. The only known example is 8778^2 + 10296^2 = 13530^2. - Andrew Howroyd, Aug 17 2018
LINKS
D. W. Ballew, R. C. Weger, Pythagorean Triples and Triangular Numbers, Fibonacci Quarterly, 17.2 (1979), 168-171.
EXAMPLE
The triangular numbers 45 and 36 are the hypotenuse and leg of a Pythagorean triple {45, 36, 27}.
PROG
(PARI) {for(i=1, 10^3, k=1; v=1; a=i*(i+1)/2; while(k<=i-1&&v, b=k*(k+1)/2; if(issquare(a*a-b*b), v=0; print1(a, ", ")); k+=1))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Antonio Roldán, Feb 28 2013
STATUS
approved