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(x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy) = t(3) = 6, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.
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%I #23 Jan 01 2024 14:05:25

%S 3,35,416,4956,59055,703703,8385380,99920856,1190664891,14188057835,

%T 169066029128,2014604291700,24006185471271,286059621363551,

%U 3408709270891340,40618451629332528,484012710281098995,5767534071743855411,68726396150645165936

%N (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy) = t(3) = 6, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.

%H Colin Barker, <a href="/A065929/b065929.txt">Table of n, a(n) for n = 0..900</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 (13,-13,1).

%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 = 3.

%F G.f.: (4*x-3) / ((1-12*x+x^2)(x-1)).

%F a(0)=3, a(1)=35, a(2)=416, a(n) = 13*a(n-1)-13*a(n-2)+a(n-3). - _Harvey P. Dale_, Mar 23 2012

%F a(n) = (2+(29-5*sqrt(35))*(6-sqrt(35))^n+(6+sqrt(35))^n*(29+5*sqrt(35)))/20. - _Colin Barker_, Mar 05 2016

%p g := (4*x-3)/(1-12*x+x^2)/(x-1): s := series(g, x, 40): for i from 0 to 30 do printf(`%d,`,coeff(s, x, i)) od: # _James A. Sellers_, Feb 11 2002

%t CoefficientList[Series[(4x-3)/((1-12x+x^2)(x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{13,-13,1},{3,35,416},30] (* _Harvey P. Dale_, Mar 23 2012 *)

%o (PARI) Vec((4*x-3)/((1-12*x+x^2)*(x-1)) + O(x^25)) \\ _Colin Barker_, Mar 05 2016

%Y Cf. A000217 (triangular numbers).

%K easy,nonn

%O 0,1

%A _Floor van Lamoen_, Nov 29 2001

%E More terms from _James A. Sellers_, Feb 11 2002