A090390 Repeatedly multiply (1,0,0) by ([1,2,2],[2,1,2],[2,2,3]); sequence gives leading entry.

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

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: A114040 A231178 A359204 * A199411 A069665 A188235

Adjacent sequences: A090387 A090388 A090389 * A090391 A090392 A090393

Keyword

easy,nonn

Author

Vim Wenders, Jan 30 2004

Status

approved