

A094429


Given the 3 X 3 matrix M = [0 1 0 / 0 0 1 / 7 14 7], a(n) = () rightmost term of M^n * [1 1 1].


10



0, 7, 42, 196, 833, 3381, 13377, 52136, 201341, 773122, 2958032, 11291903, 43042727, 163918671, 623875840, 2373568575, 9028148962, 34334213564, 130560389505, 496440779373, 1887579497489, 7176808297736, 27286630574917
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OFFSET

1,2


COMMENTS

M is derived from the Lucas polynomial: x^3  7*x^2 + 14*x  7 with a root (and eigenvalue of the matrix): 3.801377358... = (2*sin(3*Pi/7))^2, the convergent of the sequence.
From Roman Witula, Sep 29 2012: (Start)
The Berndttype sequence number 16 for the argument 2*Pi/7 (see Formula section and Crossrefs for other Berndttype sequences for the argument 2*Pi/7  for numbers from 1 to 18 without 16).
Note that all numbers of the form a(n)*7^(1  floor((n1)/3)) are integers and even a(10) and a(11) are divisible by 7^5. (End)


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1700
Roman Witula and Damian Slota, New RamanujanType Formulas and QuasiFibonacci 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(n1)  14*a(n2) + 7*a(n3).
G.f.: 7*x^2*(1x)/(1  7*x + 14*x^2  7*x^3). (End)
a(n) = c(4)*(s(1))^(2*n) + c(2)*(s(4))^(2*n) + c(1)*(s(2))^(2*n) = (1/sqrt(7))*(c(1)*(s(1))^(2*n+3) + c(2)*(s(2))^(2*n+3) + c(3)*(s(3))^(2*n+3)) = (1/sqrt(7))*(s(2)*(s(1))^(2*n+2) + s(4)*(s(2))^(2*n+2) + s(1)*(s(4))^(2*n+2)), where c(j) := 2*cos(2*Pi*j/7) and s(j) := 2*sin(2*Pi*j/7) (for the sums of the respective odd powers see A217274, see also A215493 and comments to A215494). For the proof of these formulas see WitulaSlota's paper.  Roman Witula, Jul 24 2012


EXAMPLE

a(5) = 833. M^5 * [1 1 1] = [ 42 196 833].
We have 4*a(4)  a(5) = 4*a(5)  a(6) = 7*a(2) = 49, 88*a(10) = 23*a(11), and a(3) = 6*a(2), which implies the equalities c(4)*(s(1))^6 + c(2)*(s(4))^6 + c(1)*(s(2))^6 = 6*(c(4)*(s(1))^4 + c(2)*(s(4))^4 + c(1)*(s(2))^4) and
s(2)*(s(1))^8 + s(4)*(s(2))^8 + s(1)*(s(4))^8 = 6*( s(2)*(s(1))^6 + s(4)*(s(2))^6 + s(1)*(s(4))^6).  Roman Witula, Sep 29 2012


MATHEMATICA

Table[(MatrixPower[{{0, 1, 0}, {0, 0, 1}, {7, 14, 7}}, n].{1, 1, 1})[[3]], {n, 23}] (* Robert G. Wilson v, May 08 2004 *)
LinearRecurrence[{7, 14, 7}, {0, 7, 42}, 50] (* Roman Witula, Aug 13 2012 *)


PROG

(PARI) x='x+O('x^30); concat([0], Vec(7*x^2*(1x)/(17*x+14*x^27*x^3))) \\ G. C. Greubel, May 09 2018
(PARI) a(n) = (([0, 1, 0; 0, 0, 1; 7, 14, 7]^n)*[1, 1, 1]~)[3]; \\ Michel Marcus, May 10 2018
(MAGMA) I:=[0, 7, 42]; [n le 3 select I[n] else 7*Self(n1) 14*Self(n2) +7*Self(n3): 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, A094430, A217274.
Sequence in context: A057425 A248329 A073376 * A246434 A255614 A022731
Adjacent sequences: A094426 A094427 A094428 * A094430 A094431 A094432


KEYWORD

nonn,easy


AUTHOR

Gary W. Adamson, May 02 2004


EXTENSIONS

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


STATUS

approved



