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%I #67 Apr 04 2019 10:13:05
%S 1,7,29,123,521,2207,9349,39603,167761,710647,3010349,12752043,
%T 54018521,228826127,969323029,4106118243,17393796001,73681302247,
%U 312119004989,1322157322203,5600748293801,23725150497407,100501350283429,425730551631123,1803423556807921
%N a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=7.
%C Generalized Pell equation with second term of 7.
%H T. D. Noe, <a href="/A048876/b048876.txt">Table of n, a(n) for n = 0..200</a>
%H M. Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/bicknell.pdf">A Primer on the Pell Sequence and related sequences</a>, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian Representations</a>, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A. K. Whitford, <a href="http://www.fq.math.ca/Scanned/15-1/whitford-a.pdf">Binet's Formula Generalized</a>, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,1).
%F G.f.: (1+3*x)/(1-4*x-x^2). - _Philippe Deléham_, Nov 03 2008
%F a(n) = ((1+sqrt(5))*(2+sqrt(5))^n + (1-sqrt(5))*(2-sqrt(5))^n )/2.
%F a(n) = A000032(3*n+1). - _Thomas Baruchel_, Nov 26 2003
%F From _Gary Detlefs_, Mar 06 2011: (Start)
%F a(n) = Fibonacci(3*n+7) mod Fibonacci(3*n+3), n > 0.
%F a(n) = Fibonacci(3*n+3) - Fibonacci(3*n-1). (End)
%F a(n) = A001076(n+1)+3*A001076(n). - _R. J. Mathar_, Oct 22 2013
%F a(n) = 5*F(2*n)*F(n+1) - L(n-1)*(-1)^n. - _J. M. Bergot_, Mar 22 2016
%F a(n) = Sum_{k=0..n} binomial(n,k)*5^floor((k+1)/2)*2^(n-k). - _Tony Foster III_, Sep 03 2017
%p with(combinat): a:=n->3*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # _Zerinvary Lajos_, Apr 04 2008
%t f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(2 + s)^n + (1 - s)(2 - s)^n)/2]]; (* Or *)
%t f[n_] := Fibonacci[3 n + 3] - Fibonacci[3 n - 1]; (* Or *)
%t f[n_] := Mod[ Fibonacci[3n + 7], Fibonacci[3n + 3]]; Array[f, 22, 0]
%t a[n_] := 4a[n - 1] + a[n - 2]; a[0] = 1; a[1] = 7; Array[a, 22, 0] (* Or *)
%t CoefficientList[ Series[(1 + 3x)/(1 - 4x - x^2), {x, 0, 21}], x] (* _Robert G. Wilson v_ *)
%t LinearRecurrence[{4,1},{1,7},30] (* _Harvey P. Dale_, Jun 13 2015 *)
%t Table[LucasL[3*n + 1], {n, 0, 20}] (* _Rigoberto Florez_, Apr 04 2019 *)
%o (PARI) Vec((1+3*x)/(1-4*x-x^2) + O(x^30)) \\ _Altug Alkan_, Oct 07 2015
%Y Cf. A001076, A001077, A015448, A033887.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_
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