A090390 - OEIS

%I #45 Aug 03 2024 11:18:10

%S 1,1,9,49,289,1681,9801,57121,332929,1940449,11309769,65918161,

%T 384199201,2239277041,13051463049,76069501249,443365544449,

%U 2584123765441,15061377048201,87784138523761,511643454094369,2982076586042449,17380816062160329,101302819786919521,590436102659356801

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

%C 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 Binomial transform of A086348. - _Johannes W. Meijer_, Aug 01 2010

%C 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

%D Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 60 at p. 123.

%H Harvey P. Dale, <a href="/A090390/b090390.txt">Table of n, a(n) for n = 0..1000</a>

%H Robert Munafo, <a href="http://www.mrob.com/pub/math/seq-floretion.html">Sequences Related to Floretions</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).

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

%F a(n) = A001333(n)^2

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

%F 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

%F 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

%F a(n) = (A001541(n) + (-1)^n)/2. - _R. J. Mathar_, Nov 20 2009

%F 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

%F (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

%F a(n) = A302946(n)/4. - _Eric W. Weisstein_, Apr 17 2018

%F E.g.f.: exp(-x)*(1 + exp(4*x)*cosh(2*sqrt(2)*x))/2. - _Stefano Spezia_, Aug 03 2024

%p a:= n-> (<<1|0|0>>. <<1|2|2>, <2|1|2>, <2|2|3>>^n)[1, 1]:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 17 2013

%t CoefficientList[Series[(1-4x-x^2)/((1+x)(1-6x+x^2)),{x, 0, 30}], x] (* _Harvey P. Dale_, May 20 2012 *)

%t LinearRecurrence[{5,5,-1}, {1,1,9}, 30] (* _Harvey P. Dale_, May 20 2012 *)

%t Table[(ChebyshevT[n,3]+(-1)^n)/2, {n,0,30}] (* _Eric W. Weisstein_, Apr 17 2018 *)

%t (LucasL[Range[0, 40], 2]/2)^2 (* _G. C. Greubel_, Aug 21 2022 *)

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

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

%o (PARI) Vec( (1-4*x-x^2)/((1+x)*(1-6*x+x^2)) + O(x^66) ) \\ _Joerg Arndt_, Aug 16 2013

%o (Haskell)

%o a090390 n = a090390_list !! n

%o a090390_list = 1 : 1 : 9 : zipWith (-) (map (* 5) $

%o    tail $ zipWith (+) (tail a090390_list) a090390_list) a090390_list

%o -- _Reinhard Zumkeller_, Aug 17 2013

%o (Magma) [Evaluate(DicksonFirst(n,-1),2)^2/4: n in [0..40]]; // _G. C. Greubel_, Aug 21 2022

%o (SageMath) [lucas_number2(n,2,-1)^2/4 for n in (0..40)] # _G. C. Greubel_, Aug 21 2022

%Y Cf. A000129, A001333, A001541, A079291, A086348, A095344, A123270, A302946.

%K easy,nonn

%O 0,3

%A _Vim Wenders_, Jan 30 2004