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%I #49 Sep 08 2022 08:45:01
%S 1,7,20,53,139,364,953,2495,6532,17101,44771,117212,306865,803383,
%T 2103284,5506469,14416123,37741900,98809577,258686831,677250916,
%U 1773065917,4641946835,12152774588,31816376929,83296356199,218072691668,570921718805,1494692464747
%N a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=7.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H Vincenzo Librandi, <a href="/A055267/b055267.txt">Table of n, a(n) for n = 0..1000</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F a(n) = (7*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).
%F G.f.: (1 + 4*x)/(1 - 3*x + x^2).
%F From _Rigoberto Florez_, Dec 24 2018: (Start)
%F a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n+1).
%F a(n) = 4*Fibonacci(2*n - 1) + 3*Lucas(2*n - 1). (End)
%F E.g.f.: exp(3*t/2)*( cosh(sqrt(5)*t/2) + (11/sqrt(5))*sinh(sqrt(5)*t/2) ). - _G. C. Greubel_, Jan 17 2020
%F a(n) = 4*A001906(n) + A001906(n+1). - _R. J. Mathar_, Mar 06 2022
%p with(combinat); seq(fibonacci(2*n+2) +4*fibonacci(2*n), n=0..30); # _G. C. Greubel_, Jan 17 2020
%t Table[5*Fibonacci[2n] + Fibonacci[2n+1], {n, 0, 30}]
%t Table[4*Fibonacci[2n-1] + 3*LucasL[2n-1], {n, 0, 30}] (* _Rigoberto Florez_, Dec 24 2018 *)
%t LinearRecurrence[{3,-1}, {1,7}, 30] (* _Vincenzo Librandi_, Dec 25 2018 *)
%o (PARI) Vec((1+4*x)/(1-3*x+x^2) + O(x^40)) \\ _Michel Marcus_, Sep 06 2017
%o (Magma) [5*Fibonacci(2*n) + Fibonacci(2*n+1): n in [0..30]]; // _Vincenzo Librandi_, Dec 25 2018
%o (Sage) [fibonacci(2*n+2) +4*fibonacci(2*n) for n in (0..30)] # _G. C. Greubel_, Jan 17 2020
%o (GAP) List([0..30], n-> Fibonacci(2*n+2) +4*Fibonacci(2*n) ); # _G. C. Greubel_, Jan 17 2020
%Y Cf. A000032, A000045, A054486, A054492.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 09 2000