A157668 - OEIS

%I #20 Sep 08 2022 08:45:42

%S 354,2099,5302,9963,16082,23659,32694,43187,55138,68547,83414,99739,

%T 117522,136763,157462,179619,203234,228307,254838,282827,312274,

%U 343179,375542,409363,444642,481379,519574,559227,600338,642907,686934,732419

%N a(n) = 729*n^2 - 442*n + 67.

%C The identity (531441*n^2 - 322218*n + 48842)^2 - (729*n^2 - 442*n + 67)*(19683*n - 5967)^2 = 1 can be written as A157670(n)^2 - a(n)*A157669(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A157668/b157668.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5773864&amp;tstart=0">X^2-AY^2=1</a>

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

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

%F G.f.: x*(-354 - 1037*x - 67*x^2)/(x-1)^3.

%F E.g.f.: (67 + 287*x + 729*x^2)*exp(x) - 67. - _G. C. Greubel_, Nov 17 2018

%t LinearRecurrence[{3,-3,1},{354,2099,5302},40]

%t Table[729n^2-442n+67,{n,40}] (* _Harvey P. Dale_, Dec 25 2019 *)

%o (Magma) I:=[354, 2099, 5302]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

%o (PARI) a(n) = 729*n^2 - 442*n + 67.

%o (Sage) [729*n^2 - 442*n + 67 for n in (1..40)] # _G. C. Greubel_, Nov 17 2018

%o (GAP) List([1..40], n -> 729*n^2 - 442*n + 67); # _G. C. Greubel_, Nov 17 2018

%Y Cf. A157669, A157670.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 04 2009