A244352 - OEIS

%I #16 Aug 21 2022 06:37:23

%S 0,0,4,100,1584,23548,338100,4798248,67750848,954701400,13441659268,

%T 189185124940,2662308356400,37463104912660,527155118240244,

%U 7417689205890000,104375121328998144,1468671237346368048,20665783224031936900,290789699203441908148

%N a(n) = Pell(n)^3 - Pell(n)^2, where Pell(n) is the n-th Pell number (A000129).

%H Colin Barker, <a href="/A244352/b244352.txt">Table of n, a(n) for n = 0..800</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (17,-25,-223,-79,95,-7,-1).

%F a(n) = A110272(n) - A079291(n).

%F G.f.: 4*x^2*(1+8*x-4*x^2+3*x^3) / ((1+x)*(1-6*x+x^2)*(1+2*x-x^2)*(1-14*x-x^2)).

%F a(n) = A045991(A000129(n)). - _Michel Marcus_, Jun 26 2014

%e a(3) = Pell(3)^3 - Pell(3)^2 = 5^3 - 5^2 = 100.

%t CoefficientList[Series[4*x^2*(3*x^3-4*x^2+8*x+1) / ((x+1)*(x^2-6*x+1)*(x^2-2*x-1)*(x^2+14*x-1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 26 2014 *)

%o (PARI)

%o pell(n) = round(((1+sqrt(2))^n-(1-sqrt(2))^n)/(2*sqrt(2)))

%o vector(50, n, pell(n-1)^3-pell(n-1)^2)

%o (Magma)

%o Pell:= func< n | n eq 0 select 0 else Evaluate(DicksonSecond(n-1,-1),2) >;

%o [Pell(n)^3 - Pell(n)^2: n in [0..40]]; // _G. C. Greubel_, Aug 20 2022

%o (SageMath)

%o def Pell(n): return lucas_number1(n,2,-1)

%o [Pell(n)^3 -Pell(n)^2 for n in (0..40)] # _G. C. Greubel_, Aug 20 2022

%Y Cf. A000129, A079291, A110272.

%K nonn,easy

%O 0,3

%A _Colin Barker_, Jun 26 2014