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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. 10
 1, 2, 7, 24, 80, 263, 859, 2797, 9094, 29547, 95968, 311652, 1011999, 3286051, 10669913, 34645258, 112492863, 365262680, 1186001480, 3850924183, 12503874715, 40599829957, 131826825678, 428039023363, 1389833992704, 4512762649020, 14652848312239, 47577499659779, 154483171074481, 501603705725970, 1628697001842743 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6 Index entries for linear recurrences with constant coefficients, signature (5,-6,1). FORMULA 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). G.f.: (1-3*x+3*x^2)/(1-5*x+6*x^2-x^3). a(n) = A005021(n)-3*A005021(n-1)+3*A005021(n-2). - R. J. Mathar, Aug 22 2012 EXAMPLE 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). MATHEMATICA LinearRecurrence[{5, -6, 1}, {1, 2, 7}, 50] PROG (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 (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 CROSSREFS Cf. A215512, A215695. Sequence in context: A027122 A132788 A109682 * A027124 A038765 A027126 Adjacent sequences:  A215691 A215692 A215693 * A215695 A215696 A215697 KEYWORD nonn,easy AUTHOR Roman Witula, Aug 21 2012 STATUS approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)