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 A090390 Repeatedly multiply (1,0,0) by ([1,2,2],[2,1,2],[2,2,3]); sequence gives leading entry. 17
 1, 1, 9, 49, 289, 1681, 9801, 57121, 332929, 1940449, 11309769, 65918161, 384199201, 2239277041, 13051463049, 76069501249, 443365544449, 2584123765441, 15061377048201, 87784138523761, 511643454094369, 2982076586042449, 17380816062160329, 101302819786919521, 590436102659356801 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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) Binomial transform of A086348. - Johannes W. Meijer, Aug 01 2010 All values of a(n) are squares. sqrt(a(n+1)) = A001333(n). The ratio a(n+1)/a(n) converges to 3 + 2*sqrt(2). - Richard R. Forberg, Aug 14 2013 LINKS Harvey P. Dale, Table of n, a(n) for n = 0..1000 Robert Munafo, Sequences Related to Floretions Index entries for linear recurrences with constant coefficients, signature (5, 5, -1). FORMULA G.f.: (1-4*x-x^2)/((1+x)*(1-6*x+x^2)). a(n) = A001333(n)^2 (a, b, c) = (1, 0, 0). Recursively multiply (a, b, c)*( [1, 2, 2], [2, 1, 2], [2, 2, 3] ). 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, Oct 31 2004 a(n) = (((1+sqrt(2))^(2*n) + (1-sqrt(2))^(2*n)) + 2*(-1)^n)/4 - Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 09 2005 a(n) = (A001541(n) + (-1)^n)/2. - R. J. Mathar, Nov 20 2009 a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3), with a(0)=1, a(1)=1, a(2)=9. - Harvey P. Dale, May 20 2012 (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, Nov 16 2004 a(n) = A302946(n)/4. - Eric W. Weisstein, Apr 17 2018 MAPLE a:= n-> (<<1|0|0>>. <<1|2|2>, <2|1|2>, <2|2|3>>^n)[1, 1]: seq(a(n), n=0..30); # Alois P. Heinz, Aug 17 2013 MATHEMATICA CoefficientList[Series[(1-4x-x^2)/((1+x)(1-6x+x^2)), {x, 0, 30}], x] (* Harvey P. Dale, May 20 2012 *) LinearRecurrence[{5, 5, -1}, {1, 1, 9}, 30] (* Harvey P. Dale, May 20 2012 *) Table[(ChebyshevT[n, 3]+(-1)^n)/2, {n, 0, 30}] (* Eric W. Weisstein, Apr 17 2018 *) (LucasL[Range[0, 40], 2]/2)^2 (* G. C. Greubel, Aug 21 2022 *) PROG (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]) (PARI) Vec( (1-4*x-x^2)/((1+x)*(1-6*x+x^2)) + O(x^66) ) \\ Joerg Arndt, Aug 16 2013 (Haskell) a090390 n = a090390_list !! n a090390_list = 1 : 1 : 9 : zipWith (-) (map (* 5) \$ tail \$ zipWith (+) (tail a090390_list) a090390_list) a090390_list -- Reinhard Zumkeller, Aug 17 2013 (Magma) [Evaluate(DicksonFirst(n, -1), 2)^2/4: n in [0..40]]; // G. C. Greubel, Aug 21 2022 (SageMath) [lucas_number2(n, 2, -1)^2/4 for n in (0..40)] # G. C. Greubel, Aug 21 2022 CROSSREFS Cf. A000129, A001333, A001541, A079291, A086348, A095344, A123270, A302946. Sequence in context: A123270 A114040 A231178 * A199411 A069665 A188235 Adjacent sequences: A090387 A090388 A090389 * A090391 A090392 A090393 KEYWORD easy,nonn AUTHOR Vim Wenders, Jan 30 2004 STATUS approved

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Last modified December 9 16:05 EST 2022. Contains 358701 sequences. (Running on oeis4.)