Search: id:A005386
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%I A005386 M3017
%S A005386 1,3,16,75,361,1728,8281,39675,190096,910803,4363921,20908800,100180081,
%T A005386 479991603,2299777936,11018898075,52794712441,252954664128,
%U A005386 1211978608201,5806938376875,27822713276176,133306628004003
%N A005386 Area of n-th triple of squares around a triangle.
%C A005386 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.
%D A005386 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005386 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A005386 J. C. G. Nottrot, Vierkantenkransen rond een driehoek, Pythagoras (Netherlands), 
               14 (1975-1976) 77-81.
%H A005386 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005386 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005386 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A005386 GF=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 A005386 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. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Oct 18 2004
%p A005386 A005386:=-(-1+z)/(z+1)/(z**2-5*z+1); [Conjectured by S. Plouffe in his 
               1992 dissertation.]
%p A005386 (Maple) 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); 
               [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 05 2008]
%t A005386 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 A005386 Essentially the same as A003769. First differences of A099025.
%Y A005386 Sequence in context: A004303 A005947 A003769 this_sequence A053572 A055842 
               A037773
%Y A005386 Adjacent sequences: A005383 A005384 A005385 this_sequence A005387 A005388 
               A005389
%K A005386 nonn
%O A005386 1,2
%A A005386 Jean Meeus
%E A005386 Edited by Peter J. C. Moses (mows(AT)mopar.freeserve.co.uk), Apr 23 2004
%E A005386 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004

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