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A215694 a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

%I A215694 #24 Sep 08 2022 08:46:03
%S A215694 1,2,7,24,80,263,859,2797,9094,29547,95968,311652,1011999,3286051,
%T A215694 10669913,34645258,112492863,365262680,1186001480,3850924183,
%U A215694 12503874715,40599829957,131826825678,428039023363,1389833992704,4512762649020,14652848312239,47577499659779,154483171074481,501603705725970,1628697001842743
%N A215694 a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.
%C A215694 The Berndt-type sequence number 9 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For more connections with another sequences of trigonometric nature see comments to A215512 (a(n) is equal to the sequence b(n) in these comments) and Witula-Slota's reference (Section 3). We note that a(n)=A109682(n) for n=1,2,3,4. Moreover the following summation formula hold true: sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) - 9, for every n=3,4,... - see comments to A215512.
%C A215694 The inverse binomial transform is 1,1, 4, 8, 19, 42, 95,... essentially a shifted, unsigned variant of A215112. - _R. J. Mathar_, Aug 22 2012
%H A215694 G. C. Greubel, <a href="/A215694/b215694.txt">Table of n, a(n) for n = 0..1000</a>
%H A215694 Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
%H A215694 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,1).
%F A215694 sqrt(7)*a(n) = s(4)*c(1)^(2*n) + s(1)*c(2)^(2*n) + s(2)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
%F A215694 G.f.: (1-3*x+3*x^2)/(1-5*x+6*x^2-x^3).
%F A215694 a(n) = A005021(n)-3*A005021(n-1)+3*A005021(n-2). - _R. J. Mathar_, Aug 22 2012
%e A215694 We have 10*a(3) = 3*a(4), a(0)+a(1)+3*a(2) = a(3), a(0)+a(2)+3*a(3) = a(4), a(1)+3*a(2)+3*a(4) = a(5), and a(6) = 3*a(5)+3*a(4)-a(1).
%t A215694 LinearRecurrence[{5,-6,1}, {1,2,7}, 50]
%o A215694 (PARI) Vec((1-3*x+3*x^2)/(1-5*x+6*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A215694 (Magma) I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1) - 6*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, Apr 25 2018
%Y A215694 Cf. A215512, A215695.
%K A215694 nonn,easy
%O A215694 0,2
%A A215694 _Roman Witula_, Aug 21 2012