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 A202638 y-values in the solution to x^2 - 7*y^2 = -3. 2

%I #27 Sep 08 2022 08:46:01

%S 1,2,14,31,223,494,3554,7873,56641,125474,902702,1999711,14386591,

%T 31869902,229282754,507918721,3654137473,8094829634,58236916814,

%U 129009355423,928136531551,2056054857134,14791947588002,32767868358721

%N y-values in the solution to x^2 - 7*y^2 = -3.

%C The corresponding values of x of this Pell equation are in A202637.

%H Bruno Berselli, <a href="/A202638/b202638.txt">Table of n, a(n) for n = 1..1000</a>

%H R. A. Mollin, <a href="http://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866112-5/S0025-5718-1987-0866112-5.pdf">Class Numbers of Quadratic Fields Determinet by Solvability of Diophantine Equations</a>, Mathematics of Computation Vol. 48, 1987, p. 235 (Theorem 1.1, particular case).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,16,0,-1).

%F G.f.: x*(1-x)*(1+3*x+x^2)/(1-16*x^2+x^4).

%F a(n) = a(-n+1) = ((7+2*sqrt(7)*(-1)^n)*(8-3*sqrt(7))^floor(n/2)+(7-2*sqrt(7)*(-1)^n)*(8+3*sqrt(7))^floor(n/2))/14.

%F a(2n)+a(2n-1) = A202637(2n)-A202637(2n-1).

%t LinearRecurrence[{0,16,0,-1}, {1, 2, 14, 31}, 24]

%o (PARI) a=vector(24); a[1]=1; a[2]=2; a[3]=14; a[4]=31; for(i=5, #a, a[i]=16*a[i-2]-a[i-4]); a

%o (Magma) m:=24; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1+3*x+x^2)/(1-16*x^2+x^4)));

%o (Maxima) makelist(expand(((7+2*sqrt(7)*(-1)^n)*(8-3*sqrt(7))^floor(n/2)+(7-2*sqrt(7)*(-1)^n)*(8+3*sqrt(7))^floor(n/2))/14), n, 1, 24);

%K nonn,easy

%O 1,2

%A _Bruno Berselli_, Dec 22 2011

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Last modified July 19 22:34 EDT 2024. Contains 374441 sequences. (Running on oeis4.)