A094430 a(n) is the rightmost term of M^n * [1 0 0], where M is the 3 X 3 matrix [0 1 0 / 0 0 1 / 7 -14 7].

7, 49, 245, 1078, 4459, 17836, 69972, 271313, 1044435, 4002467, 15294370, 58337097, 222255768, 846131608, 3219700183, 12247849145, 46582062709, 177142452214, 673583231587, 2561162729076, 9737971026812, 37024601601729

Offset

1,1

Comments

In A094429 the multiplier is [1 1 1] instead of [1 0 0]. The matrix M is derived from the 3rd-order Lucas polynomial x^3 - 7x^2 + 14x - 7, with a convergent of the series = 3.801937735... = (2 sin 3*Pi/7)^2; (an eigenvalue of the matrix and a root of the polynomial).

From Roman Witula, Sep 29 2012: (Start)

This sequence is the Berndt-type sequence number 17 for the argument 2*Pi/7 (see Formula section and Crossrefs for other Berndt-type sequences for the argument 2*Pi/7 - for numbers from 1 to 18 without 17).

Note that all numbers of the form a(n)*7^(-floor((n+4)/3)) are integers. (End)

Links

G. C. Greubel, Table of n, a(n) for n = 1..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 (7,-14,7).

Formula

From Colin Barker, Jun 19 2012: (Start)

a(n) = 7*a(n-1)-14*a(n-2)+7*a(n-3).

G.f.: 7*x/(1-7*x+14*x^2-7*x^3). (End)

-a(n) = s(2)*s(1)^(2*n+3) + s(4)*s(2)^(2*n+3) + s(1)*s(4)^(2*n+3), where s(j) := 2*sin(2*Pi*j/7); for the proof see A215494 and the Witula-Slota paper. This formula and the respective recurrence also give a(0)=a(-1)=0. - Roman Witula, Aug 13 2012

Example

a(4) = 1078 since M^4 * [1 0 0] = [49 245 1078] = [a(2), a(3), a(4)].
We have a(2)=7*a(1), a(3)=5*a(2), 22*a(3)=5*a(4), and a(6)=4*a(5), which implies s(2)*s(1)^15 + s(4)*s(2)^15 + s(1)*s(4)^15 = 4*(s(2)*s(1)^13 + s(4)*s(2)^13 + s(1)*s(4)^13). - Roman Witula, Sep 29 2012

Mathematica

Table[(MatrixPower[{{0, 1, 0}, {0, 0, 1}, {7, -14, 7}}, n].{1, 0, 0})[[3]], {n, 22}] (* Robert G. Wilson v, May 08 2004 *)
Join[{7}, LinearRecurrence[{7, -14, 7}, {49, 245, 1078}, 50]] (* Roman Witula, Aug 13 2012 *)(* corrected by G. C. Greubel, May 09 2018 *)

Prog

(PARI) x='x+O('x^30); Vec(7*x/(1-7*x+14*x^2-7*x^3)) \\ G. C. Greubel, May 09 2018
(PARI) a(n) = (([0, 1, 0; 0, 0, 1; 7, -14, 7]^n)*[1, 0, 0]~)[3]; \\ Michel Marcus, May 10 2018
(Magma) I:=[49, 245, 1078]; [7] cat [n le 3 select I[n] else 7*Self(n-1) -14*Self(n-2) + 7*Self(n-3): n in [1..30]]; // G. C. Greubel, May 09 2018

Crossrefs

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A217274.

Sequence in context: A375610 A362392 A224150 * A188561 A225013 A228459

Adjacent sequences: A094427 A094428 A094429 * A094431 A094432 A094433

Keyword

nonn,easy

Author

Gary W. Adamson, May 02 2004

Extensions

More terms from Robert G. Wilson v, May 08 2004

Name edited by Michel Marcus, May 10 2018

Status

approved