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Search: id:A106731
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| A106731 |
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First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-2],[1,4]] and v is the column vector [0,1]. |
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+0
1
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| 0, -2, -8, -28, -96, -328, -1120, -3824, -13056, -44576, -152192, -519616, -1774080, -6057088, -20680192, -70606592, -241065984, -823050752, -2810071040, -9594182656, -32756588544, -111837988864, -381838778368, -1303679135744, -4451038986240, -15196797673472
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Real Pisot roots (the eigenvalues of M): 2-sqrt(2)= 0.585786, 2+sqrt(2)=3.41421. a(n)=-2*A007070(n-1) for n>=1.
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FORMULA
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a(n)=first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, -2], [1, 4]] and v[0] is the column vector [0,1]. G.f.=-2x/(1-4x+2x^2). a(n)=4a(n-1)-2a(n-2); a(0)=0, a(1)=-2.
a(n)=-(1/2)*sqrt(2)*[2+sqrt(2)]^n+(1/2)*[2-sqrt(2)]^n*sqrt(2), with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Oct 07 2008]
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MAPLE
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a[0]:=0: a[1]:=-2: for n from 2 to 27 do a[n]:=4*a[n-1]-2*a[n-2] od: seq(a[n], n=0..27);
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MATHEMATICA
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M = {{0, -2}, {1, 4}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}]
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CROSSREFS
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Cf. A060995, A007070.
Sequence in context: A087431 A090426 A060995 this_sequence A066796 A104934 A056711
Adjacent sequences: A106728 A106729 A106730 this_sequence A106732 A106733 A106734
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KEYWORD
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sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 30 2005
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 30 2006
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