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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).

%I A224749 #44 Oct 19 2024 15:57:32
%S A224749 0,1,3,15,69,321,1491,6921,32139,149229,692919,3217437,14939559,
%T A224749 69369021,322101927,1495619397,6944625855,32246056989,149728468167,
%U A224749 695235829509,3228196110975,14989518216045,69600993441975,323179052074101,1500620817813327,6967849012498557,32353889326768359
%N 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).
%C A224749 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
%D A224749 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).
%H A224749 G. C. Greubel, <a href="/A224749/b224749.txt">Table of n, a(n) for n = 0..1000</a>
%H A224749 Pierre de la Harpe, <a href="http://images-archive.math.cnrs.fr/Vauban-pour-les-cochons-comme.html">Vauban pour les cochons comme Fibonacci pour les lapins</a>, Images des Mathématiques, CNRS, 2013.
%H A224749 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, 6, 6, 6, 6).
%F A224749 G.f.: x/(1-3*x-6*x^2-6*x^3-6*x^4-6*x^5). - _Philippe Deléham_, Apr 17 2013
%p A224749 f:=proc(n) option remember;
%p A224749 if n <= 0 then 0 elif n=1 then 1 else
%p A224749 3*f(n-1)+6*f(n-2)+6*f(n-3)+6*f(n-4)+6*f(n-5); fi; end;
%p A224749 [seq(f(n),n=0..30)];
%t A224749 LinearRecurrence[{3, 6, 6, 6, 6}, {0, 1, 3, 15, 69}, 40] (* _T. D. Noe_, Apr 17 2013 *)
%t A224749 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 *)
%o A224749 (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
%o A224749 (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
%Y A224749 Cf. A000045 (Fibonacci), A000930 (Narayana).
%K A224749 nonn,easy
%O A224749 0,3
%A A224749 Pierre de la Harpe, Apr 17 2013