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%I #16 Feb 06 2023 07:32:26
%S 1,6,25,111,488,2149,9461,41654,183389,807403,3554736,15650361,
%T 68903513,303360038,1335596817,5880203831,25888648920,113979406525,
%U 501814720109,2209329044566,9726966211957,42824708216851,188543436246752,830096195208753,3654646945111665
%N Sequence generated from a symmetric matrix.
%C A100295 is generated from M^n * [1, 0, 0].
%C Limit_{n -> oo} a(n)/a(n-1) tends to 4.4026788295...a root of the characteristic polynomial of M, x^3 - 4*x^2 - 2*x + 1 and an eigenvalue of M.
%H G. C. Greubel, <a href="/A100296/b100296.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,-1).
%F a(n) = rightmost term in M^n * [1,1,1], where M = [3,2,1; 2,1,0; 1,0,0].
%F a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3).
%F G.f.: x*(1+2*x-x^2)/(1-4*x-2*x^2+x^3). - _Colin Barker_, May 25 2013
%e a(5) = 488 since M^5 * [1, 1, 1] = [2149, 1263, 488]. 2149 = a(6).
%e a(8) = 4*a(7) + 2*a(6) - a(5) =41654 = 4*9461 + 2*2149 - 488.
%p a:= n-> (<<3|2|1>, <2|1|0>, <1|0|0>>^n. <<1, 1, 1>>)[3,1]:
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Feb 06 2023
%t LinearRecurrence[{4,2,-1}, {1,6,25}, 40] (* _G. C. Greubel_, Feb 05 2023 *)
%o (Magma) I:=[1,6,25]; [n le 3 select I[n] else 4*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // _G. C. Greubel_, Feb 05 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A100296
%o if (n<3): return (1,1,6)[n]
%o else: return 4*a(n-1) + 2*a(n-2) - a(n-3)
%o [a(n) for n in range(1,41)] # _G. C. Greubel_, Feb 05 2023
%Y Cf. A100295, A100297.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Nov 11 2004
%E More terms from _Colin Barker_, May 25 2013
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