

A213188


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


1



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
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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), 168171.


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<=i1&&v, b=k*(k+1)/2; if(issquare(a*ab*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



