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y-values in the solution to x^2 - 20*y^2 = 176.
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%I #27 Nov 03 2024 06:16:07

%S 1,2,5,7,14,19,37,50,97,131,254,343,665,898,1741,2351,4558,6155,11933,

%T 16114,31241,42187,81790,110447,214129,289154,560597,757015,1467662,

%U 1981891,3842389,5188658,10059505,13584083,26336126,35563591,68948873,93106690

%N y-values in the solution to x^2 - 20*y^2 = 176.

%C Also y-values in the solution of x^2-5*y^2=44.

%H Vincenzo Librandi, <a href="/A228208/b228208.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).

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

%F a(n) = 3*a(n-2)-a(n-4).

%F Let h(n) = hypergeom([(1 - n)/2, (n + 1) mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 2. - _Peter Luschny_, Sep 03 2019

%t CoefficientList[Series[(x + 1) (x^2 + x + 1) / ((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 17 2013 *)

%o (PARI) Vec(x*(x+1)*(x^2+x+1)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

%o (Magma) I:=[1,2,5,7,14]; [n le 4 select I[n] else 3*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Aug 17 2013

%Y Cf. A228207, A054486 (bisection).

%K nonn,easy

%O 1,2

%A _Colin Barker_, Aug 16 2013