Search: id:A090390
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%I A090390
%S A090390 1,1,9,49,289,1681,9801,57121,332929,1940449,11309769,65918161,
%T A090390 384199201,2239277041,13051463049,76069501249,443365544449,
%U A090390 2584123765441,15061377048201,87784138523761
%N A090390 Repeatedly multiply (1,0,0) by ([1,2,2],[2,1,2],[2,2,3]); sequence gives 
               leading entry.
%C A090390 The values of a and b in (a,b,c)*A give all (positive integer) solutions 
               to Pell equation a^2 - 2*b^2 = -1; the values of c are A000129(2n)
%C A090390 (a(n)) = tesseq(- .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' 
               + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e), apart from initial 
               term. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Nov 16 2004
%F A090390 (a, b, c) = (1, 0, 0). Recursivly multiply (a, b, c)*( [1, 2, 2], [2, 
               1, 2], [2, 2, 3] ).
%F A090390 G.f.: (1-4x-x^2)/((1+x)(1-6x+x^2)).
%F A090390 M^n * [ 1 1 1] = [a(n+1) q a(n)], where M = the 3 X 3 matrix [4 4 1 / 
               2 1 0 / 1 0 0]. E.g. M^5 * [1 1 1] = [9801 4059 1681] where 9801 
               = a(6), 1681 = a(5). Similarly, M^n * [1 0 0] generates A079291 (Pell 
               number squares). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 31 
               2004
%F A090390 (((1+sqrt(2))^(2*n)+(1-sqrt(2))^(2*n))+2*(-1)^n)/4 - Lambert Klasen (lambert.klasen(AT)gmx.net), 
               Oct 09 2005
%F A090390 a(n) = (A001541(n)+(-1)^n)/2. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 20 2009]
%o A090390 (Perl) use Math::Matrix; use Math::BigInt; $a = new Math::Matrix ([ 1, 
               2, 2], [ 2, 1, 2], [ 2, 2, 3]); $p = new Math::Matrix ([1, 0, 0]); 
               $p->print(); for ($i=1; $i<20;$i++) { $p = $p->multiply($a); $p->
               print(); }
%o A090390 (PARI) a(n)=polcoeff((1-4*x-x^2)/((1+x)*(1-6*x+x^2))+x*O(x^n),n)
%o A090390 (PARI) a(n)=if(n<0,0,([1,2,2;2,1,2;2,2,3]^n)[1,1])
%Y A090390 a(n) = A001333(n)^2. Cf. A000129.
%Y A090390 Cf. A079291.
%Y A090390 Sequence in context: A012231 A123270 A114040 this_sequence A069665 A066558 
               A135625
%Y A090390 Adjacent sequences: A090387 A090388 A090389 this_sequence A090391 A090392 
               A090393
%K A090390 easy,nonn,new
%O A090390 0,3
%A A090390 Vim Wenders (vim(AT)gmx.li), Jan 30 2004

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