A156735 - OEIS

%I #28 Jan 30 2024 19:29:42

%S 9801,114243,332929,665859,1113033,1674451,2350113,3140019,4044169,

%T 5062563,6195201,7442083,8803209,10278579,11868193,13572051,15390153,

%U 17322499,19369089,21529923,23805001,26194323,28697889,31315699

%N a(n) = 57122*n^2 + 47320*n + 9801.

%C The identity (57122*n^2 +47320*n +9801)^2 - (169*n^2 +140*n +29)*(4394*n +1820)^2 = 1 can be written as a(n)^2 - A156640(n)*A156636(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A156735/b156735.txt">Table of n, a(n) for n = 0..10000</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) for n>2.

%F G.f.: (9801 + 84840*x + 19603*x^2)/(1 - x)^3. -  _Vincenzo Librandi_, May 03 2014

%F E.g.f.: (9801 +104442*x +57122*x^2)*exp(x). - _G. C. Greubel_, Feb 28 2021

%p A156735:= n-> 57122*n^2 + 47320*n + 9801; seq(A156735(n), n=0..50); # _G. C. Greubel_, Feb 28 2021

%t LinearRecurrence[{3,-3,1},{9801,114243,332929},50]

%t CoefficientList[Series[(9801 +84840x +19603x^2)/(1-x)^3, {x, 0, 60}], x] (* _Vincenzo Librandi_, May 03 2014 *)

%t Table[57122n^2+47320n+9801,{n,0,30}] (* _Harvey P. Dale_, Jan 30 2024 *)

%o (Magma) I:=[9801, 114243, 332929]; [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)= 57122*n^2+47320*n+9801 \\ _Charles R Greathouse IV_, Dec 23 2011

%o (Sage) [57122*n^2 + 47320*n + 9801 for n in (0..50)] # _G. C. Greubel_, Feb 28 2021

%Y Cf. A156636, A156640, A156718.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Feb 15 2009

%E Edited by _Charles R Greathouse IV_, Jul 25 2010