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Expansion of (5-x^2)/((1+x)*(1-6*x+x^2)).
2

%I #56 Mar 11 2024 23:13:31

%S 5,25,149,865,5045,29401,171365,998785,5821349,33929305,197754485,

%T 1152597601,6717831125,39154389145,228208503749,1330096633345,

%U 7752371296325,45184131144601,263352415571285,1534930362283105,8946229758127349,52142448186480985

%N Expansion of (5-x^2)/((1+x)*(1-6*x+x^2)).

%C Old name was: Relates the squares of Pell numbers with the squares of the numerators of continued fraction convergents to sqrt(2).

%C Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[(j' + k' + 'ii')*('j + 'k + 'ii')] - _Creighton Dement_, Aug 16 2005

%H G. C. Greubel, <a href="/A098212/b098212.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).

%F G.f.: (5-x^2)/((1+x)*(1-6*x+x^2)).

%F a(n) = 4*A079291(n+1) + A090390(n+1) = 4(A000129(n+1))^2 + (A001333(n+1))^2.

%F a(n) + a(n+1) = A075848(n+2) - A075848(n+1).

%F a(n) = A001541(n+1) + 2*A079291(n+1). - _Creighton Dement_, Oct 26 2004

%F a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3), a(0) = 5, a(1) = 25, a(2) = 149. - _Robert G. Wilson v_, Nov 05 2004

%F 2*a(n) = (-1)^n + 3*A001541(n+1). - _R. J. Mathar_, Sep 11 2019

%t a[0]= 5; a[1]= 25; a[2]= 149; a[n_]:= a[n]= 5 a[n-1] + 5 a[n-2] - a[n-3]; Table[ a[n], {n,0,40}] (* _Robert G. Wilson v_, Nov 05 2004 *)

%t CoefficientList[Series[(5-x^2)/((1+x)(1-6x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{5,5,-1},{5,25,149},40] (* _Harvey P. Dale_, Jun 09 2011 *)

%o (PARI) Vec((5-x^2)/((1+x)*(1-6*x+x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%o (Magma) I:=[5,25,149]; [n le 3 select I[n] else 5*Self(n-1)+5*Self(n-2)-Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jul 26 2015

%o (SageMath)

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

%o [4*Pell(n+1)^2 +(Pell(n+1) +Pell(n))^2 for n in (0..40)] # _G. C. Greubel_, Aug 20 2022

%Y Cf. A000129, A001333, A001541, A079291, A075848, A090390.

%K nonn,easy

%O 0,1

%A _Creighton Dement_, Oct 25 2004