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%I #19 Sep 08 2022 08:45:18
%S 6,20,34,56,126,204,330,736,1190,1924,4290,6936,11214,25004,40426,
%T 65360,145734,235620,380946,849400,1373294,2220316,4950666,8004144,
%U 12940950,28854596,46651570,75425384,168176910,271905276,439611354,980206864
%N Values of y in x^2 - 289 = 2*y^2.
%H G. C. Greubel, <a href="/A106528/b106528.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F a(3n) = 34*A001109(n).
%F a(3n+1) = A001109(n+2) + 4*A001109(n+1) - 3*A001109(n) + 4*A001109(n-1).
%F a(3n+2) = 4*A001109(n+2) - 3*A001109(n+1) + 4*A001109(n) + A001109(n-1).
%F From _Colin Barker_, Mar 29 2012: (Start)
%F a(n) = 6*a(n-3) - a(n-6).
%F G.f.: 2*x*(3 +10*x +17*x^2 +10*x^3 +3*x^4)/(1 -6*x^3 +x^6). (End)
%e a(9) = 1190, 34*A001109(3) = 34*35 = 1190;
%e a(10) = 1924, A001109(5) +4*A001109(4) -3*A001109(3) +4*A001109(2) = 1189 + 4*204 - 3*35 + 4*6 = 1924;
%e a(11) =4290, 4*A001109(5) -3*A001109(4) +4*A001109(3) +A001109(2) = 4*1189 - 3*204 + 4*35 + 6 = 4290; also 2*4290^2 = A106527(11)^2 - 289 = 6067^2 - 289.
%t LinearRecurrence[{0,0,6,0,0,-1}, {6,20,34,56,126,204}, 40] (* _G. C. Greubel_, Aug 18 2021 *)
%o (Magma) I:=[6,20,34,56,126,204]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // _G. C. Greubel_, Aug 18 2021
%o (Sage)
%o def A106528_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 2*x*(3 +10*x +17*x^2 +10*x^3 +3*x^4)/(1 -6*x^3 +x^6) ).list()
%o a=A106528_list(41); a[1:] # _G. C. Greubel_, Aug 18 2021
%Y Cf. A001109, A106527.
%K nonn
%O 1,1
%A Andras Erszegi (erszegi.andras(AT)chello.hu), May 09 2005
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