0,3

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)

(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

(a, b, c) = (1, 0, 0). Recursivly multiply (a, b, c)*( [1, 2, 2], [2, 1, 2], [2, 2, 3] ).

G.f.: (1-4x-x^2)/((1+x)(1-6x+x^2)).

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

(((1+sqrt(2))^(2*n)+(1-sqrt(2))^(2*n))+2*(-1)^n)/4 - Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 09 2005

(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(); }

(PARI) a(n)=polcoeff((1-4*x-x^2)/((1+x)*(1-6*x+x^2))+x*O(x^n), n)

(PARI) a(n)=if(n<0, 0, ([1, 2, 2; 2, 1, 2; 2, 2, 3]^n)[1, 1])

a(n) = A001333(n)^2. Cf. A000129.

Cf. A079291.

Sequence in context: A012231 A123270 A114040 this_sequence A069665 A066558 A135625

Adjacent sequences: A090387 A090388 A090389 this_sequence A090391 A090392 A090393

easy,nonn

Vim Wenders (vim(AT)gmx.li), Jan 30 2004