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%I #12 Oct 25 2023 11:47:38
%S 1,6,27,114,468,1896,7632,30624,122688,491136,1965312,7862784,
%T 31454208,125822976,503304192,2013241344,8053014528,32212156416,
%U 128848822272,515395682304,2061583515648,8246335635456,32985345687552
%N a(n) = 6*a(n-1) - 8*a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.
%C Binomial transform of A037480; second binomial transform of A133600.
%C First differences of A080960.
%H Vincenzo Librandi, <a href="/A171475/b171475.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F a(n) = 3*(5*4^n - 2*2^n)/8 for n > 0.
%F G.f.: (1-x)*(1+x)/((1-2*x)*(1-4*x)).
%F E.g.f.: (1/8)*(-1 - 6*exp(2*x) + 15*exp(4*x)). - _G. C. Greubel_, Dec 02 2021
%t Table[If[n==0, 1, 3*(5*4^n - 2*2^n)/8],{n,0,30}] (* _G. C. Greubel_, Dec 02 2021 *)
%t LinearRecurrence[{6,-8},{1,6,27},30] (* _Harvey P. Dale_, Oct 25 2023 *)
%o (PARI) {m=21; v=concat([1, 6, 27], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]); v}
%o (Magma) I:=[6,27]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 8*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 02 2021
%o (Sage) [1]+[3*(5*4^n - 2*2^n)/8 for n in (1..30)] # _G. C. Greubel_, Dec 02 2021
%Y Cf. A037480 ((5*3^n +(-1)^n -6)/8), A133600 (row sums of triangle A133599), A080960 (third binomial transform of A010685).
%K nonn,easy
%O 0,2
%A _Klaus Brockhaus_, Dec 09 2009
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