A156676 - OEIS

%I #44 Oct 19 2024 13:45:06

%S 6,43,242,603,1126,1811,2658,3667,4838,6171,7666,9323,11142,13123,

%T 15266,17571,20038,22667,25458,28411,31526,34803,38242,41843,45606,

%U 49531,53618,57867,62278,66851,71586,76483,81542,86763,92146,97691,103398,109267,115298,121491

%N a(n) = 81*n^2 - 44*n + 6.

%C The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - a(n)*A156772(n)^2 = 1 for n > 0.

%C For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-3; {1, 1, 3, 1, 9n-4, 1, 3, 1, 1, 18n-6}]. - _Magus K. Chu_, Sep 13 2022

%H Vincenzo Librandi, <a href="/A156676/b156676.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).

%F G.f.: (6 + 25*x + 131*x^2)/(1-x)^3.

%F a(n) = A000290(A017245(n-1)) - A017137(n-1). - _Reinhard Zumkeller_, Jul 13 2010

%F E.g.f.: (6 + 37*x + 81*x^2)*exp(x). - _Elmo R. Oliveira_, Oct 19 2024

%p A156676:=n->81*n^2-44*n+6: seq(A156676(n), n=0..100); # _Wesley Ivan Hurt_, Apr 26 2017

%t LinearRecurrence[{3,-3,1},{6,43,242},40]

%t Table[81n^2-44n+6,{n,0,40}] (* _Harvey P. Dale_, Oct 29 2019 *)

%o (Magma) [81*n^2 - 44*n + 6: n in [0..40] ];

%o (PARI) a(n)=81*n^2-44*n+6 \\ _Charles R Greathouse IV_, Dec 23 2011

%Y Cf. A000290, A017137, A017245, A156772, A156774.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Feb 15 2009

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