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A005386 Area of n-th triple of squares around a triangle.
(Formerly M3017)

%I M3017

%S 1,3,16,75,361,1728,8281,39675,190096,910803,4363921,20908800,

%T 100180081,479991603,2299777936,11018898075,52794712441,252954664128,

%U 1211978608201,5806938376875,27822713276176,133306628004003

%N Area of n-th triple of squares around a triangle.

%C a(n)*(-1)^(n+1) is the r=-3 member of the r-family of sequences S_r(n), n>=1, defined in A092184 where more information can be found.

%C The sequence is the case P1 = 3, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Apr 03 2014

%D J. C. G. Nottrot, Vierkantenkransen rond een driehoek, Pythagoras (Netherlands), 14 (1975-1976) 77-81.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H J. Meeus, <a href="/A005386/a005386.pdf">Letter to N. J. A. Sloane with attachment, Mar 1975</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.pdf">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume

%F G.f.: x*(1-x)/(x^3-4*x^2-4*x+1), a(n)=4*(a(n-1)+a(n-2))-a(n-3), a(1)=1, a(2)=3, a(3)=16

%F a(n) = (2/7)*(T(n, 5/2)-(-1)^n) with twice Chebyshev's polynomials of the first kind evaluated at x=5/2: 2*T(n, 5/2)=A003501(n)= ((5+sqrt(21))^n + (5-sqrt(21))^n)/2^n. _Wolfdieter Lang_, Oct 18 2004

%F a(2n) = A003690(n). a(2n+1) = A004253(n)^2. - Alexander Evnin, Mar 11 2012

%F From _Peter Bala_, Apr 03 2014: (Start)

%F a(n)= |U(n-1,sqrt(3)*i/2)|^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.

%F a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.

%F See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

%p A005386:=-(-1+z)/(z+1)/(z**2-5*z+1); [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]

%p a:= n-> (Matrix([[0,1,3]]). Matrix(3, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,4,-1][i] else 0 fi)^(n))[1,1] ; seq (a(n), n=1..22); # _Alois P. Heinz_, Aug 05 2008

%t a[n_]:=Module[{n1=1, n2=0}, Do[{n1, n2}={Sqrt[3]*n1+n2, n1}, {n-1}];n1^2] a[n_]:=Round[((5+Sqrt[21])/2)^n/7] (CoefficientList[Series[{(x/(1-x*(Sqrt[3]+x)))}, {x, 0, 20}], x])^2 CoefficientList[Series[{x*(1-x)/(x^3-4*x^2-4*x+1)}, {x, 0, 20}], x]

%Y Essentially the same as A003769. First differences of A099025. A100047.

%K nonn

%O 1,2

%A Jean Meeus

%E Edited by _Peter J. C. Moses_, Apr 23 2004

%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004

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Last modified February 16 18:53 EST 2019. Contains 320165 sequences. (Running on oeis4.)