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 A268409 a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5. 0

%I

%S 3,5,26,114,508,2260,10056,44744,199088,885840,3941536,17537824,

%T 78034368,347213120,1544921216,6874111104,30586286848,136093369600,

%U 605546052096,2694370947584,11988575894528,53343045473280,237349333682176,1056083425675264

%N a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5.

%C In general, the ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m, is (k - (k*r - m)*x)/(1 - r*x - s*x^2). This recurrence gives the closed form b(n) = (2^(-n - 1)*((k*r - 2*m)*(r - sqrt(r^2 + 4*s))^n + (2*m - k*r)*(sqrt(r^2 + 4*s) + r)^n + k*sqrt(r^2 + 4*s)*(r - sqrt(r^2 + 4*s))^n + k*sqrt(r^2 + 4*s)*(sqrt(r^2 + 4*s) + r)^n))/sqrt(r^2 + 4*s).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,2).

%F G.f.: (3 - 7*x)/(1 - 4*x - 2*x^2).

%F a(n) = ((18 + sqrt(6))*(2 - sqrt(6))^n - (sqrt(6) - 18)*(2 + sqrt(6))^n)/12.

%F Lim_{n -> infinity} a(n + 1)/a(n) = 2 + sqrt(6) = A176213.

%F a(n) = 3*A090017(n+1) -7*A090017(n). - _R. J. Mathar_, Mar 12 2017

%t RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == 4 a[n - 1] + 2 a[n - 2]}, a, {n, 23}]

%t LinearRecurrence[{4, 2}, {3, 5}, 24]

%t Table[((18 + Sqrt[6]) (2 - Sqrt[6])^n - (Sqrt[6] - 18) (2 + Sqrt[6])^n)/12, {n, 0, 23}]

%o (MAGMA) [n le 2 select 2*n+1 else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 04 2016

%o (PARI) Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ _Michel Marcus_, Feb 04 2016

%Y Cf. A021001, A084059, A090017, A107979, A164549, A176213, A189741.

%K nonn,easy

%O 0,1

%A _Ilya Gutkovskiy_, Feb 04 2016

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Last modified October 15 05:56 EDT 2018. Contains 316202 sequences. (Running on oeis4.)