A141385 a(n) = 7*a(n-1) - 9*a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.

3, 7, 31, 157, 827, 4407, 23563, 126105, 675075, 3614143, 19349431, 103593805, 554625899, 2969386479, 15897666067, 85113810057, 455687062275, 2439682811479, 13061709929935, 69930511268509, 374397872321627

Offset: 0

Author

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

Keywords

• nonn
• easy
• nice

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.

Examples

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

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).

Crossrefs

Cf. A141221, A141384.

Programs

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

Maple

m:=30; S:=series( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 30 2021

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 *)

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

Sage

def A141385_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3) ).list()
A141385_list(40) # G. C. Greubel, Mar 30 2021

Formula

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

Extensions

New definition by Bruno Berselli, Oct 22 2012