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%I #24 Sep 16 2015 09:18:00
%S 1,1,19,28,406,721,8722,17983,188209,438697,4078270,10530100,88714549,
%T 249679990,1936716229,5864281633,42418800739,136706927896,
%U 931844786950,3167777090989,20525689021222,73046232419419,453213909082585
%N Expansion of (1+x-2*x^2)/(1-21*x^2-7*x^3).
%C We have a(n)=A(n;3), where A(n;d), n=1,2,..., d in C denote one of the three quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. The remaining two "conjugate" sequences B(n;3) and C(n;3) can be obtained from the following system of recurrence equations: A(0;3)=1, B(0;3)=C(0;3)=0, A(n+1;3)=A(n;3)+6*B(n;3)-3*C(n;3), B(n+1;3)=3*A(n;3)+B(n;3), C(n+1;3)=3*B(n;3)-2*C(n;3). These sequences are also connected by very intriguing convolution type relations (in some sense limiting in the nature) - see identities (3.53-55) and the identities (3.47-49) (the last ones for the value d=3) in the cited paper. We note that each of the three numbers a(3*n), a(3*n+1) and a(3*n+2) is divided by 7^n for every n=0,1,..., which follows easily from recurrence relation for the sequence a(n). - _Roman Witula_, Aug 11 2012
%H Harvey P. Dale, <a href="/A121458/b121458.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 21, 7).
%F a(0)=a(1)=1, a(2)=19, a(n+1) = 21*a(n-1)+7*a(n-2) for n>=2.
%F a(n) = (1/7)*((s(2))^2*(1+3*c(1))^n + (s(4))^2*(1+3*c(2))^n + (s(1))^2*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper). - _Roman Witula_, Aug 11 2012
%t CoefficientList[Series[(1 + x - 2*x^2)/(1 - 21*x^2 - 7*x^3), {x, 0, 200}], x] (* _Stefan Steinerberger_, Sep 11 2006 *)
%t LinearRecurrence[{0,21,7},{1,1,19},30] (* _Harvey P. Dale_, May 19 2012 *)
%o (PARI) Vec((1+x-2*x^2)/(1-21*x^2-7*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%K nonn,easy
%O 0,3
%A _Philippe Deléham_, Sep 07 2006
%E More terms from _Stefan Steinerberger_, Sep 11 2006
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