A213188 Triangular numbers that are hypotenuse and a leg of a Pythagorean triple.

10, 45, 136, 325, 435, 595, 630, 666, 780, 1225, 2080, 2145, 3321, 5050, 5565, 5886, 6216, 7381, 7503, 9316, 10440, 11026, 11175, 12246, 13530, 14196, 14365, 14535, 15753, 16653, 18915, 19306, 24310, 25425, 32896, 33670, 39060, 41905, 42195, 49141, 50721, 52650

Offset: 1

Author

Antonio Roldán, Feb 28 2013

Keywords

• nonn

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

Examples

			The triangular numbers 45 and 36 are the hypotenuse and leg of a Pythagorean triple {45, 36, 27}.
		

Links

• D. W. Ballew, R. C. Weger, Pythagorean Triples and Triangular Numbers, Fibonacci Quarterly, 17.2 (1979), 168-171.

Crossrefs

Cf. A126200, A213189, A217843.

Programs

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))}