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

0,3

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).

Formula

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A304276 A217451 A213451 * A122558 A110211 A167874

Adjacent sequences: A224746 A224747 A224748 * A224750 A224751 A224752

Keyword

nonn,easy

Author

Pierre de la Harpe, Apr 17 2013

Status

approved