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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 19:22 EST 2019. Contains 329809 sequences. (Running on oeis4.)