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A261397 a(n) = 3^n*Fibonacci(n).

%I A261397 #34 Jan 05 2025 19:51:40
%S A261397 0,3,9,54,243,1215,5832,28431,137781,669222,3247695,15766083,76527504,
%T A261397 371477259,1803179313,8752833270,42487113627,206236840311,
%U A261397 1001094543576,4859415193527,23588096472765,114499026160038,555789946734999,2697861075645339,13095692747551008,63567827923461075
%N A261397 a(n) = 3^n*Fibonacci(n).
%H A261397 Colin Barker, <a href="/A261397/b261397.txt">Table of n, a(n) for n = 0..1000</a>
%H A261397 Tom Edgar, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/54-1/Edgar10292015.pdf">Extending Some Fibonacci-Lucas Relations</a>, Fib. Quarterly, 54 (2016), 79.
%H A261397 D. Marques, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.7.683">A new Fibonacci-Lucas relation</a>, Amer. Math. Monthly, 122 (2015), 683.
%H A261397 Ivica Martinjak, <a href="http://arxiv.org/abs/1508.04949">Complementary Families of the Fibonacci-Lucas Relations</a>, arXiv:1508.04949 [math.CO], 2015.
%H A261397 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,9).
%F A261397 a(n) = 3*a(n-1) + 9*a(n-2), a(0)=0, a(1)=3. - _G. C. Greubel_, Aug 21 2015
%F A261397 G.f.: 3*x / (1 - 3*x - 9*x^2). - _G. C. Greubel_, Aug 21 2015
%F A261397 E.g.f.: (1/(phi - 1/phi))*(e^(3*phi*x) - e^(3*x/phi)), where 2*phi = 1 + sqrt(5). - _G. C. Greubel_, Aug 21 2015
%t A261397 RecurrenceTable[{a[0]== 0, a[1]== 3, a[n]== 3*a[n-1]  + 9*a[n-2]}, a, {n,50}] (* _G. C. Greubel_, Aug 21 2015 *)
%t A261397 LinearRecurrence[{3, 9}, {0, 3}, 30] (* _Vincenzo Librandi_, Aug 21 2015 *)
%o A261397 (PARI) vector(30, n, n--; 3^n*fibonacci(n)) \\ _Michel Marcus_, Aug 21 2015
%o A261397 (PARI) concat(0, Vec(-3*x/(9*x^2+3*x-1) + O(x^30))) \\ _Colin Barker_, Sep 01 2015
%o A261397 (Magma) [3^n*Fibonacci(n): n in [0..30]]; // _Vincenzo Librandi_, Aug 21 2015
%Y A261397 Cf. A000045, A103435.
%K A261397 nonn,easy
%O A261397 0,2
%A A261397 _N. J. A. Sloane_, Aug 18 2015