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

1,1

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

Table of n, a(n) for n=1..42.

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

Cf. A126200, A213189, A217843.

Sequence in context: A022605 A211032 A179095 * A037270 A027800 A005714

Adjacent sequences: A213185 A213186 A213187 * A213189 A213190 A213191

Keyword

nonn

Author

Antonio Roldán, Feb 28 2013

Status

approved