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a(n) = 531441*n^2 - 740664*n + 258065.
3

%I #18 Sep 08 2022 08:45:42

%S 48842,902501,2819042,5798465,9840770,14945957,21114026,28344977,

%T 36638810,45995525,56415122,67897601,80442962,94051205,108722330,

%U 124456337,141253226,159112997,178035650,198021185,219069602,241180901

%N a(n) = 531441*n^2 - 740664*n + 258065.

%C The identity (531441*n^2 - 740664*n + 258065)^2 - (729*n^2 - 1016*n + 354)*(19683*n - 13716)^2 = 1 can be written as a(n)^2 - A157665(n)*A157666(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A157667/b157667.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*(48842 + 755975*x + 258065*x^2)/(1-x)^3.

%F E.g.f.: (258065 - 209223*x + 531441*x^2)*exp(x) - 258065. - _G. C. Greubel_, Nov 17 2018

%t LinearRecurrence[{3,-3,1},{48842,902501,2819042},40]

%o (Magma) I:=[48842, 902501, 2819042]; [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) = 531441*n^2 - 740664*n + 258065.

%o (Sage) [531441*n^2 - 740664*n + 258065 for n in (1..40)] # _G. C. Greubel_, Nov 17 2018

%o (GAP) List([1..40], n -> 531441*n^2 - 740664*n + 258065); # _G. C. Greubel_, Nov 17 2018

%Y Cf. A157665, A157666.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 04 2009