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%I #18 Sep 08 2022 08:45:40
%S 7,88,555,3276,19135,111568,650307,3790308,22091575,128759176,
%T 750463515,4374021948,25493668207,148587987328,866034255795,
%U 5047617547476,29419671029095,171470408627128,999402780733707,5824946275775148
%N a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 7, a(1) = 88, a(2) = 555.
%C lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).
%H G. C. Greubel, <a href="/A155465/b155465.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) = 7, a(1) = 88.
%F a(n) = ((31+25*sqrt(2))*(3+2*sqrt(2))^n + (31-25*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.
%F G.f.: (7+39*x-12*x^2)/((1-x)*(1-6*x+x^2)).
%F a(n) = (3*A002203(2*n+3) + 10*A002203(2*n+1) - 34)/4. - _G. C. Greubel_, Aug 21 2018
%t LinearRecurrence[{7,-7,1},{7,88,555},30] (* _Harvey P. Dale_, Apr 29 2012 *)
%t Table[(3*LucasL[2*n+3,2] + 10*LucasL[2*n+1,2] - 34)/4, {n, 0, 50}] (* _G. C. Greubel_, Aug 21 2018 *)
%o (PARI) {m=20; v=concat([7, 88, 555], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}
%o (Magma) I:=[7,88,555]; [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 Second trisection of A118120. Cf. A001652.
%Y Cf. A155464, A155466, 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
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