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A213188 Triangular numbers that are hypotenuse and a leg of a Pythagorean triple. 1

%I

%S 10,45,136,325,435,595,630,666,780,1225,2080,2145,3321,5050,5565,5886,

%T 6216,7381,7503,9316,10440,11026,11175,12246,13530,14196,14365,14535,

%U 15753,16653,18915,19306,24310,25425,32896,33670,39060,41905,42195,49141,50721,52650

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

%C 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.

%C 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

%H D. W. Ballew, R. C. Weger, <a href="https://www.fq.math.ca/Scanned/17-2/ballew.pdf">Pythagorean Triples and Triangular Numbers</a>, Fibonacci Quarterly, 17.2 (1979), 168-171.

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

%o (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))}

%Y Cf. A126200, A213189, A217843.

%K nonn

%O 1,1

%A _Antonio Roldán_, Feb 28 2013

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