A005386 - OEIS

%I M3017 #61 Nov 17 2022 07:20:52

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

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

%U 1211978608201,5806938376875,27822713276176,133306628004003,638710426743841,3060245505715200

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

%H G. C. Greubel, <a href="/A005386/b005386.txt">Table of n, a(n) for n = 1..1000</a>

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

%H J. C. G. Nottrot, <a href="https://pyth.eu/uploads/user/ArchiefPDF/Pyth14-4.pdf">Vierkantenkransen rond een driehoek</a>, Pythagoras (Netherlands), 14 (1975-1976) 77-81.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%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="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-1).

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

%F a(n) = 4*a(n-1) + 4*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..25); # _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];

%t Table[a[n], {n,30}]

%t a[n_]:= Round[((5+Sqrt[21])/2)^n/7]; Table[a[n], {n, 30}]

%t Rest@(CoefficientList[Series[x/(1-x*(Sqrt[3]+x)), {x, 0, 30}], x])^2

%t Abs[ChebyshevU[Range[1,40]-1, I*Sqrt[3]/2]]^2 (* _G. C. Greubel_, Nov 16 2022 *)

%o (Magma) I:=[1, 3, 16]; [n le 3 select I[n] else 4*Self(n-1) +4*Self(n-2) -Self(n-3): n in [1..41]]; // _G. C. Greubel_, Nov 16 2022

%o (SageMath)

%o def A005386(n): return abs(chebyshev_U(n-1, i*sqrt(3)/2))^2

%o [A005386(n) for n in range(1,40)] # _G. C. Greubel_, Nov 16 2022

%Y Essentially the same as A003769.

%Y First differences of A099025.

%Y Cf. A100047.

%K nonn,easy

%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