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A224419 Numbers n such that triangular(n) + triangular(2*n) is a square.

%I A224419 #55 Nov 14 2024 10:23:48
%S A224419 0,1,25,216,1849,36481,311904,2666689,52606009,449765784,3845364121,
%T A224419 75857828929,648561949056,5545012396225,109386936710041,
%U A224419 935225880773400,7995904029992761,157735886878050625,1348595071513294176,11530088066237165569,227455039491212291641,1944673157896289428824,16626378995609962758169,327990009210441246496129
%N A224419 Numbers n such that triangular(n) + triangular(2*n) is a square.
%C A224419 8 of the first 10 terms are of the form x^y. The two exceptions are a(7) = 311904 = 2^5 * 3^3 * 19^2 and a(10) = 449765784 = 2^3 * 3^5 * 13^2 * 37^2.
%C A224419 The corresponding squares are given by A075873(2*n-1)^2. E.g., triangular(a(10)) + triangular(2*a(10)) = 711142146^2 = A075873(19)^2.
%C A224419 Locations of squares in A147875, equivalent to solving the Diophantine equation n*(5*n+3)=2*s^2. - _R. J. Mathar_, Apr 19 2013
%H A224419 Max Alekseyev, <a href="/A224419/b224419.txt">Table of n, a(n) for n = 1..1000</a>
%H A224419 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1442,-1442,0,-1,1).
%F A224419 a(n) = (A228209(2*n-1) - 3) / 10. - _Max Alekseyev_, Sep 04 2013
%F A224419 G.f.: x^2*(x+1)*(x^4 + 23*x^3 + 168*x^2 + 23*x + 1) / (x^6 - 1442*x^3 + 1) / (1-x). - _Max Alekseyev_, Sep 04 2013
%t A224419 LinearRecurrence[{1,0,1442,-1442,0,-1,1},{0,1,25,216,1849,36481,311904},30]  (* _Harvey P. Dale_, Jan 23 2015 *)
%o A224419 (Python)
%o A224419 import math
%o A224419 for i in range(1<<30):
%o A224419         s = i*(i+1)/2 + i*(2*i+1)
%o A224419         t = int(math.sqrt(s))
%o A224419         if s == t*t:  print(i)
%Y A224419 Cf. A000217, A075873.
%Y A224419 Cf. A220186 (numbers n such that triangular(2*n) - triangular(n) is a square).
%K A224419 nonn
%O A224419 1,3
%A A224419 _Alex Ratushnyak_, Apr 18 2013
%E A224419 Terms a(11) onward from _Max Alekseyev_, Sep 04 2013