%I #12 Sep 08 2022 08:45:40

%S 28,207,1248,7315,42676,248775,1450008,8451307,49257868,287095935,

%T 1673317776,9752810755,56843546788,331308470007,1931007273288,

%U 11254735169755,65597403745276,382329687301935,2228380720066368

%N a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.

%C lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

%H G. C. Greubel, <a href="/A155466/b155466.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).

%F a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 28, a(1) = 207.

%F a(n) = ((73+53*sqrt(2))*(3+2*sqrt(2))^n + (73-53*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.

%F G.f.: (28+11*x-5*x^2)/((1-x)*(1-6*x+x^2)).

%F a(n) = (10*A002203(2*n+3) + 3*A002203(2*n+1) - 34)/4. - _G. C. Greubel_, Aug 21 2018

%t Table[(10*LucasL[2*n+3,2] + 3*LucasL[2*n+1, 2] -34)/4, {n, 0, 50}] (* or *) LinearRecurrence[{7,-7,1}, {28, 207, 1248}, 50] (* _G. C. Greubel_, Aug 21 2018 *)

%o (PARI) {m=19; v=concat([28, 207, 1248], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

%o (Magma) I:=[28, 207, 1248]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // _G. C. Greubel_, Aug 21 2018

%Y Third trisection of A118120. Cf. A001652.

%Y Cf. A155464, A155465, A156035 (decimal expansion of 3+2*sqrt(2)).

%K nonn,easy

%O 0,1

%A _Klaus Brockhaus_, Jan 30 2009

%E Comment and recursion formula added, cross-references edited by _Klaus Brockhaus_, Sep 23 2009