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A141385 a(n) = 7*a(n-1)-9*a(n-2)+a(n-3) with a(0)=3, a(1)=7, a(2)=31. 2
3, 7, 31, 157, 827, 4407, 23563, 126105, 675075, 3614143, 19349431, 103593805, 554625899, 2969386479, 15897666067, 85113810057, 455687062275, 2439682811479, 13061709929935, 69930511268509, 374397872321627 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The old definition given for this sequence was "A sequence obeying a third-order linear recurrence".

Ruling out finitely many exceptional terms, this sequence differs by a constant from several related enumerations with a slightly more complicated structure (fourth-order linear recurrence):

For n>0, A141221(n)=a(n)-1. For n>2, A141384(n)=a(n)+1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

G. P. Michon, Silent Prisms: A Screaming Game for Short-Sighted People.

Index entries for linear recurrences with constant coefficients, signature (7,-9,1).

FORMULA

G.f.: (3-14x+9x^2)/(1-7x+9x^2-x^3).

a(n+3) = 7*a(n+2)-9*a(n+1)+a(n).

a(n) = A^n + B^n + C^n, where, putting u = atan(sqrt(5319)/73), we have:

A = 5.3538557854308282... = (7+2*sqrt(22)*cos(u/3))/3

B = 1.5235479602692093... = (7-sqrt(22)*cos(u/3)+sqrt(66)*sin(u/3))/3

C = 0.1225962542999624... = (7-sqrt(22)*cos(u/3)-sqrt(66)*sin(u/3))/3

EXAMPLE

a(0) = 3 = A^0+B^0+C^0, a(1) = 7 = A+B+C.

MATHEMATICA

LinearRecurrence[{7, -9, 1}, {3, 7, 31}, 40] (* Harvey P. Dale, May 25 2011 *)

CoefficientList[Series[(3 - 14x + 9x^2)/(1 - 7x + 9x^2 - x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2012 *)

PROG

(MAGMA) I:=[3, 7, 31]; [n le 3 select I[n] else 7*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -9, 7]^n*[3; 7; 31])[1, 1] \\ Charles R Greathouse IV, Feb 10 2017

CROSSREFS

Cf. A141221, A141384.

Sequence in context: A000644 A015459 A115083 * A059296 A123332 A051342

Adjacent sequences:  A141382 A141383 A141384 * A141386 A141387 A141388

KEYWORD

nonn,easy,nice

AUTHOR

Gerard P. Michon, Jul 02 2008, Jul 23 2008

EXTENSIONS

New definition by Bruno Berselli, Oct 22 2012

STATUS

approved

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Last modified April 28 18:26 EDT 2017. Contains 285579 sequences.