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A065928 (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy)) = t(2) = 3, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2. 1

%I

%S 2,11,63,366,2132,12425,72417,422076,2460038,14338151,83568867,

%T 487075050,2838881432,16546213541,96438399813,562084185336,

%U 3276066712202,19094316087875,111289829815047,648644662802406,3780578146999388,22034824219193921,128428367168164137

%N (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy)) = t(2) = 3, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.

%H Vincenzo Librandi, <a href="/A065928/b065928.txt">Table of n, a(n) for n = 0..200</a>

%H J.-P. Ehrmann et al., <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/POLYA002.html">Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).

%F a(n) = 2*t(m)*a(n-1)-a(n-2)-1, a(0) = m, a(1) = m^3+m^2-1 with m = 2.

%F G.f.: (3*x-2)/((1-6*x+x^2)*(x-1)).

%F a(0)=2, a(1)=11, a(2)=63, a(n)=7*a(n-1)-7*a(n-2)+a(n-3). - _Harvey P. Dale_, Nov 06 2011

%F a(n) = (4+(14-11*sqrt(2))*(3-2*sqrt(2))^n+(3+2*sqrt(2))^n*(14+11*sqrt(2)))/16. - _Colin Barker_, Mar 05 2016

%t CoefficientList[Series[(3x-2)/((1-6x+x^2)(x-1)),{x,0,20}],x] (* or *) LinearRecurrence[{7,-7,1},{2,11,63},20] (* _Harvey P. Dale_, Nov 06 2011 *)

%o (MAGMA) I:=[2,11,63]; [n le 3 select I[n] else 7*Self(n-1)-7*Self(n-2)+Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Nov 13 2011

%o (PARI) Vec((3*x-2)/((1-6*x+x^2)*(x-1)) + O(x^40)) \\ _Colin Barker_, Mar 05 2016

%K nonn,easy

%O 0,1

%A _Floor van Lamoen_, Nov 29 2001

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Last modified May 31 02:12 EDT 2020. Contains 334747 sequences. (Running on oeis4.)