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a(n) = 2*n^2 + 14*n + 1.
7

%I #41 Oct 27 2024 16:08:24

%S 1,17,37,61,89,121,157,197,241,289,341,397,457,521,589,661,737,817,

%T 901,989,1081,1177,1277,1381,1489,1601,1717,1837,1961,2089,2221,2357,

%U 2497,2641,2789,2941,3097,3257,3421,3589,3761,3937,4117,4301,4489,4681,4877,5077

%N a(n) = 2*n^2 + 14*n + 1.

%C The defining formula can be regarded as an approximation and simplification of the expansion/propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition/concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by _Pierre Gayet_]

%H Pierre Gayet, <a href="/A161549/b161549.txt">Table of n, a(n) for n = 0..9999</a>

%H Pierre Gayet, <a href="/A162316/a162316.gif">Note et Compte rendu</a> (gif version).

%H Pierre Gayet, <a href="/A162316/a162316.pdf">Note et Compte Rendu</a> (pdf version).

%H Pierre Gayet, <a href="/A162316/a162316_1.txt">98 séquences générées ... par la formule générale indiquée</a>.

%H Claude Monet, <a href="https://web.archive.org/web/20060428022333/http://lycees.ac-rouen.fr/bruyeres/jardin/Nymphea.html">Nymphéas</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + 4*n + 12 (with a(0)=1). - _Vincenzo Librandi_, Nov 30 2010

%F G.f.: (1 + 14*x - 11*x^2)/(1-x)^3. - _Vincenzo Librandi_, Nov 08 2014

%F From _Elmo R. Oliveira_, Oct 25 2024: (Start)

%F E.g.f.: (1 + 16*x + 2*x^2)*exp(x).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t lst={}; Do[a=2*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jun 13 2009 *)

%t CoefficientList[Series[(1 + 14 x - 11 x^2) / (1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 08 2014 *)

%t Table[2n^2+14n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,17,37},50] (* _Harvey P. Dale_, Jul 14 2018 *)

%o (Magma) [ 2*n^2+14*n+1: n in [0..50] ];

%o (PARI) Vec((1+14*x-11*x^2)/(1-x)^3 + O(x^100)) \\ _Colin Barker_, Nov 08 2014

%Y Cf. A161532, A161587, A161617, A161935, A162316.

%K easy,nonn

%O 0,2

%A _Pierre Gayet_, Jun 13 2009

%E More terms from _Vladimir Joseph Stephan Orlovsky_, Jun 13 2009