|
%I #16 Sep 08 2022 08:45:40
%S 0,51,340,2023,11832,69003,402220,2344351,13663920,79639203,464171332,
%T 2705388823,15768161640,91903581051,535653324700,3122016367183,
%U 18196444878432,106056652903443,618143472542260,3602804182350151
%N a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.
%C lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).
%H Harvey P. Dale, <a href="/A155464/b155464.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) = 0, a(1) = 51.
%F a(n) = ((1+sqrt(2))*(3+2*sqrt(2))^n + (1-sqrt(2))*(3-2*sqrt(2))^n -2)*(17/4).
%F G.f.: 17*x*(3-x)/((1-x)*(1-6*x+x^2)).
%F a(n) = 17*(A002203(2*n+1) - 2)/4. - _G. C. Greubel_, Aug 21 2018
%t LinearRecurrence[{7,-7,1},{0,51,340},30] (* _Harvey P. Dale_, Jun 10 2013 *)
%t Table[17*(LucasL[2*n+1,2] - 2)/4, {n, 0, 50}] (* _G. C. Greubel_, Aug 21 2018 *)
%o (PARI) {m=20; v=concat([0, 51, 340], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}
%o (Magma) I:=[0,51,340]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, Aug 21 2018
%Y First trisection of A118120. Equals 17*A001652.
%Y Cf. A155465, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).
%K nonn,easy
%O 0,2
%A _Klaus Brockhaus_, Jan 30 2009
%E Comment and recursion formula added, cross-references edited by _Klaus Brockhaus_, Sep 23 2009
|
