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A215492 a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

%I A215492 #34 Sep 08 2022 08:46:03
%S A215492 0,3,6,63,147,1365,3528,29694,83643,648270,1964361,14199171,45789471,
%T A215492 311933118,1060973088,6871121775,24463966674,151720368891,
%U A215492 561841152579,3357375513429,12860706786396,74437773850062,293576471108319,1653218198356074,6686170310225133
%N A215492 a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=0, a(1)=3, and a(2)=6.
%C A215492 We have a(n)=B(n;3), where B(n;d), n=1,2,..., d \in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and C(n;3) are discussed in A121458 and A215484 respectively. Similarly as in A121458 we deduce that each of the following elements a(3*n), a(3*n+1), a(3*n+2) is divided by 3*7^n for every n=0,1,... .
%H A215492 G. C. Greubel, <a href="/A215492/b215492.txt">Table of n, a(n) for n = 0..1000</a>
%H A215492 Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H A215492 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 21, 7).
%F A215492 a(n) = (1/7)*((c(1)-c(4))*(1+3*c(1))^n + (c(2)-c(1))*(1+3*c(2))^n + (c(4)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper).
%F A215492 G.f.: (3*x+6*x^2)/(1-21*x^2-7*x^3).
%t A215492 LinearRecurrence[{0,21,7}, {0,3,6}, 50]
%t A215492 CoefficientList[Series[(3 x + 6 x^2)/(1 - 21 x^2 - 7 x^3), {x, 0, 33}], x] (* _Vincenzo Librandi_, Sep 18 2015 *)
%o A215492 (PARI) concat(0,Vec((3+6*x)/(1-21*x^2-7*x^3)+O(x^99))) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A215492 (Magma) I:=[0,3,6]; [n le 3 select I[n] else 21*Self(n-2)+7*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Sep 18 2015
%Y A215492 Cf. A121458, A215484, A121449, A085810, A215404, A077998, A006054, A033304, A052975, A094789, A005021, A121442, A121458.
%K A215492 nonn,easy
%O A215492 0,2
%A A215492 _Roman Witula_, Aug 13 2012