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%I #38 Sep 08 2022 08:45:15
%S 2,5,13,32,78,189,457,1104,2666,6437,15541,37520,90582,218685,527953,
%T 1274592,3077138,7428869,17934877,43298624,104532126,252362877,
%U 609257881,1470878640,3551015162,8572908965,20696833093,49966575152,120629983398,291226541949
%N a(n) = (1/2) * (5*P(n+1) + P(n) - 1), where P(k) are the Pell numbers A000129.
%H Colin Barker, <a href="/A098586/b098586.txt">Table of n, a(n) for n = 0..1000</a>
%H Hermann Stamm-Wilbrandt, <a href="/A098586/a098586.svg">4 interlaced bisections</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).
%F a(n) = 3*a(n-1) - a(n-2) - a(n-3) with a(0)=2, a(1)=5, a(2)=13. - _Hermann Stamm-Wilbrandt_, Aug 26 2014
%F G.f.: (2-x)/((1-x)*(1-2*x-x^2)). - _Robert Israel_, Aug 26 2014
%F a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>5. - _Hermann Stamm-Wilbrandt_, Aug 27 2014
%F a(2*n-1) = A006451(2*n), for n>0. - _Hermann Stamm-Wilbrandt_, Aug 27 2014
%F a(2*n) = A124124(2*n+2). - _Hermann Stamm-Wilbrandt_, Aug 27 2014
%F a(n) = (-2+(5-3*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(5+3*sqrt(2)))/4. - _Colin Barker_, Mar 16 2016
%p A:= LREtools[REtoproc](a(n) = 3*a(n-1) - a(n-2) - a(n-3), a(n), {a(0)=2, a(1)=5, a(2)=13}):
%p seq(A(n),n=0..100); # _Robert Israel_, Aug 26 2014
%t LinearRecurrence[{3, -1, -1}, {2, 5, 13}, 28] (* _Hermann Stamm-Wilbrandt_, Aug 26 2014 *)
%t CoefficientList[Series[(2-x)/((1-x)*(1-2*x-x^2)), {x,0,50}], x] (* _G. C. Greubel_, Feb 03 2018
%o (PARI) Vec((2-x)/((1-x)*(1-2*x-x^2)) + O(x^50)) \\ _Colin Barker_, Mar 16 2016
%o (Magma) I:=[2,5,13]; [n le 3 select I[n] else 3*Self(n-1) - Self(n-2) - Self(n-3): n in [1..30]]; // _G. C. Greubel_, Feb 03 2018
%Y Cf. A006451, A124124.
%K nonn,easy
%O 0,1
%A _Creighton Dement_, Oct 03 2004
%E Formula supplied by _Thomas Baruchel_, Oct 03 2004
%E More terms from _Emeric Deutsch_, Nov 17 2004
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