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A284874 List of pairs (a,d) of coprime integers a>0, d>=0 such that a*(a+d)*(a+2*d) is a square, ordered by the squares.

%I A284874 #17 Apr 06 2017 21:10:09
%S A284874 1,0,1,24,18,7,1,840,49,120,50,119,49,240,128,161,98,527,289,336,800,
%T A284874 41,162,1519,288,1081,529,840,1,28560,49,5280,961,720,289,2520,242,
%U A284874 3479,49,9360,512,3713,529,3696,1568,1241,338,6887,2401,1320,2178,2047
%N A284874 List of pairs (a,d) of coprime integers a>0, d>=0 such that a*(a+d)*(a+2*d) is a square, ordered by the squares.
%C A284874 This is a 2-column table read by rows. For each row a,d the product a*(a+d)*(a+2*d) is a square. The rows are ordered by those products.
%C A284874 The main entry for this sequence is A284666, formed by the triples a, a+d, a+2*d. The square roots of the products a*(a+d)*(a+2*d) form A284876.
%C A284874 For a=1 the d values 0, 24, 840, 28560, ... are A078522.
%H A284874 Giovanni Resta, <a href="/A284874/b284874.txt">Table of n, a(n) for n = 1..832</a>
%F A284874 a(2*k+1) = A284666(3*k+1) and a(2*k+2) = A284666(3*k+2)-A284666(3*k+1) and a(2*k+1)*[a(2*k+1)+a(2*k+2)]*[a(2*k+1)+2*a(2*k+2)] = A284876(k+1)^2 for k>=0.
%e A284874 gcd(18,7)=1 and 18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2, so 18,7 is in the sequence.
%t A284874 nn = 50000; t = {};
%t A284874 p[a_, d_] := a (a + d) (a + 2 d); Do[
%t A284874 If[p[a, d] <= 2 nn^2 && GCD[a, d] == 1 && IntegerQ[Sqrt[p[a, d]]],
%t A284874   AppendTo[t, {a, d}]], {a, 1, nn}, {d, 0, nn}];
%t A284874 Sort[t, p[#1[[1]], #1[[2]]] < p[#2[[1]], #2[[2]]] &] // Flatten
%Y A284874 Cf. A078522, A284666, A284876.
%K A284874 nonn,tabf
%O A284874 1,4
%A A284874 _Jonathan Sondow_, Apr 04 2017
%E A284874 a(37)-a(52) from _Giovanni Resta_, Apr 06 2017