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

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

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