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A225786 Numbers k such that oblong(2*k) + oblong(k) is a square, where oblong(k) = A002378(k) = k*(k+1).

%I A225786 #29 Jan 12 2018 10:10:56
%S A225786 0,48,15552,5007792,1612493568,519217921200,167186558132928,
%T A225786 53833552500881712,17334236718725778432,5581570389877199773488,
%U A225786 1797248331303739601284800,578708381109414274413932208,186342301468900092621684886272
%N A225786 Numbers k such that oblong(2*k) + oblong(k) is a square, where oblong(k) = A002378(k) = k*(k+1).
%C A225786 Numbers k such that k*(5*k+3) is a perfect square. Apparently a(n) = 323*a(n-1) -323*a(n-2) +a(n-3). - _R. J. Mathar_, May 18 2013
%H A225786 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A225786 G.f.: 48*x*(1+x)/((1-x)*(1-322*x+x^2)). - _Bruno Berselli_, May 18 2013
%F A225786 a(n) = (3/20)*((2-sqrt(5))^(4n-4)+(2+sqrt(5))^(4n-4)-2). - _Bruno Berselli_, May 18 2013
%e A225786 48*49 + 96*97 = 108^2, so 48 is in the sequence.
%t A225786 LinearRecurrence[{323, -323, 1}, {0, 48, 15552}, 15] (* _Bruno Berselli_, May 18 2013 *)
%o A225786 (C)
%o A225786 #include <stdio.h>
%o A225786 #include <math.h>
%o A225786 int main() {
%o A225786   unsigned long long i, s, t;
%o A225786   for (i = 0; i< (1ULL<<31); i++) {
%o A225786     s = 2*i*(2*i+1) + i*(i+1);
%o A225786     t = sqrt(s);
%o A225786     if (s==t*t) printf("%llu, ", i);
%o A225786   }
%o A225786   return 0;
%o A225786 }
%Y A225786 Cf. A002378.
%Y A225786 Cf. A098301 (numbers n such that oblong(2*n) - oblong(n) is a square).
%Y A225786 Cf. A224419 (triangular(2*n) + triangular(n) is a square).
%Y A225786 Cf. A220186 (triangular(2*n) - triangular(n) is a square).
%Y A225786 Cf. A225785 (oblong(2*n) + oblong(n) is an oblong number).
%K A225786 nonn,easy
%O A225786 1,2
%A A225786 _Alex Ratushnyak_, May 16 2013
%E A225786 a(6) from _Ralf Stephan_, May 17 2013
%E A225786 More terms from _Bruno Berselli_, May 18 2013