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%I #39 Sep 08 2022 08:46:03
%S 0,0,9,0,189,63,3969,2646,83790,83349,1778112,2336859,37923795,
%T 61520823,812757708,1557403848,17498557629,38394784764,378371537145,
%U 928780383447,8214565773393,22152988812402,179007343925382,522714725474193,3914225144119836,12230060642435727,85857731104835907,284230849499989119
%N a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=a(1)=0, a(2)=9.
%C We have a(n)=C(n;3), where C(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 B(n;3) are discussed in A121458 and A215492, respectively. Similarly as in A121458, we deduce that each of the elements a(3*n), a(3*n+1), a(3*n+2) are divisible by 9*7^n for every n=0,1,... . Some additional facts connecting all three sequences a(n), A121458, and A215492 are given in the comments to A121458.
%H G. C. Greubel, <a href="/A215484/b215484.txt">Table of n, a(n) for n = 0..1000</a>
%H R. Witula, D. Slota and A. 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 21, 7).
%F a(n) = (1/7)*((c(2)-c(4))*(1+3*c(1))^n + (c(4)-c(1))*(1+3*c(2))^n + (c(1)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see formula (3.17) for d=3 in the Witula-Slota-Warzynski paper).
%F G.f.: 9*x^2/(1-21*x^2-7*x^3).
%e We have a(5)=7*a(2), a(4)=21*a(2), a(4)=3*a(5), a(6)=21*a(4), a(7)=14*a(4), 3*a(7)=2*a(6), a(8)-a(9)=7*a(5), a(9)=21*a(6), 2*a(9)=63*a(7), a(12)=455*a(9) - especially the values and the relations connecting with a(8) and a(9) are very attractive.
%t LinearRecurrence[{0,21,7}, {0,0,9}, 50]
%t CoefficientList[Series[9x^2/(1-21x^2-7x^3),{x,0,30}],x] (* _Harvey P. Dale_, Jul 06 2021 *)
%o (PARI) x='x+O('x^30); concat([0,0], Vec(9*x^2/(1-21*x^2-7*x^3))) \\ _G. C. Greubel_, Apr 19 2018
%o (Magma) I:=[0,0,9]; [n le 3 select I[n] else 21*Self(n-2) +7*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Apr 19 2018
%K nonn,easy
%O 0,3
%A _Roman Witula_, Aug 13 2012
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