A224749 Vauban's sequence: a(n)=0 if n<=0, a(1)=1; thereafter a(n) = 3*a(n-1) + 6*a(n-2) + 6*a(n-3) + 6*a(n-4) + 6*a(n-5).

0, 1, 3, 15, 69, 321, 1491, 6921, 32139, 149229, 692919, 3217437, 14939559, 69369021, 322101927, 1495619397, 6944625855, 32246056989, 149728468167, 695235829509, 3228196110975, 14989518216045, 69600993441975, 323179052074101, 1500620817813327, 6967849012498557, 32353889326768359

Offset: 0

Author

Pierre de la Harpe, Apr 17 2013

Keywords

• nonn
• easy

Comments

In his essay "La Cochonnerie ou calcul estimatif...", French military engineer Vauban (1633-1707) writes about this Fibonacci-like sequence for the year-by-year growth of pigs. - Charles R Greathouse IV, Sep 16 2015

References

• Sébastien Le Prestre de Vauban, La cochonnerie ou calcul estimatif pour connaître jusqu'où peut aller la production d'une truie pendant dix années de temps (1699).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Pierre de la Harpe, Vauban pour les cochons comme Fibonacci pour les lapins, Images des Mathématiques, CNRS, 2013.
• Index entries for linear recurrences with constant coefficients, signature (3, 6, 6, 6, 6).

Crossrefs

Cf. A000045 (Fibonacci), A000930 (Narayana).

Programs

Magma

I:=[0,1,3,15,69]; [n le 5 select I[n] else 3*Self(n-1)+6*Self(n-2)+6*Self(n-3)+6*Self(n-4)+6*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Sep 17 2015

Maple

f:=proc(n) option remember;
if n <= 0 then 0 elif n=1 then 1 else
3*f(n-1)+6*f(n-2)+6*f(n-3)+6*f(n-4)+6*f(n-5); fi; end;
[seq(f(n),n=0..30)];

Mathematica

LinearRecurrence[{3, 6, 6, 6, 6}, {0, 1, 3, 15, 69}, 40] (* T. D. Noe, Apr 17 2013 *)
CoefficientList[Series[x/(1 - 3 x - 6 x^2 - 6 x^3 - 6 x^4 - 6 x^5), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 17 2015 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 6,6,6,6,3]^n*[0;1;3;15;69])[1,1] \\ Charles R Greathouse IV, Sep 16 2015

Formula

G.f.: x/(1-3*x-6*x^2-6*x^3-6*x^4-6*x^5). - Philippe Deléham, Apr 17 2013