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A209084 a(n) = 2*a(n-1) + 4*a(n-2) with n>1, a(0)=0, a(1)=4.

%I A209084 #44 Jun 13 2025 01:09:37
%S A209084 0,4,8,32,96,320,1024,3328,10752,34816,112640,364544,1179648,3817472,
%T A209084 12353536,39976960,129368064,418643968,1354760192,4384096256,
%U A209084 14187233280,45910851584,148570636288,480784678912,1555851902976,5034842521600,16293092655104
%N A209084 a(n) = 2*a(n-1) + 4*a(n-2) with n>1, a(0)=0, a(1)=4.
%C A209084 a(n)/A063727(n) are convergents for A134972.
%C A209084 Abs(Sum_{i=0..n} C(n,n-i)*a(i)-(sqrt(5)-1)* A033887(n))->0. - _Seiichi Kirikami_, Jan 20 2016
%D A209084 E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.
%H A209084 Bruno Berselli, <a href="/A209084/b209084.txt">Table of n, a(n) for n = 0..500</a>
%H A209084 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,4).
%F A209084 a(n) = (2/sqrt(5))*((1+sqrt(5))^n-(1-sqrt(5))^n).
%F A209084 G.f.: 4*x/(1-2*x-4*x^2). - _Bruno Berselli_, Mar 08 2012
%F A209084 a(n) = 4*A085449(n) = 2*A103435(n). - _Bruno Berselli_, Mar 08 2012
%F A209084 Sum_{n>=1} 1/a(n) = (1/4) * A269991. - _Amiram Eldar_, Feb 01 2021
%t A209084 RecurrenceTable[{a[n]==2*a[n-1]+4*a[n-2], a[0]==0, a[1]==4}, a, {n, 30}]
%t A209084 LinearRecurrence[{2, 4}, {0, 4}, 40] (* _Vincenzo Librandi_, Jan 16 2016 *)
%o A209084 (PARI) concat(0, Vec(4*x/(1-2*x-4*x^2) + O(x^40))) \\ _Michel Marcus_, Jan 16 2016
%o A209084 (Magma) I:=[0,4]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jan 16 2016
%Y A209084 Cf. A063727, A033887, A085449, A103435, A134972, A269991.
%Y A209084 Cf. A086344 (this sequence with signs).
%K A209084 nonn,easy
%O A209084 0,2
%A A209084 _Seiichi Kirikami_, Mar 06 2012