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A002533 a(n) = 2*a(n-1) + 5*a(n-2).
(Formerly M4369 N1834)
29

%I M4369 N1834

%S 1,1,7,19,73,241,847,2899,10033,34561,119287,411379,1419193,4895281,

%T 16886527,58249459,200931553,693110401,2390878567,8247309139,

%U 28449011113,98134567921,338514191407,1167701222419,4027973401873

%N a(n) = 2*a(n-1) + 5*a(n-2).

%C The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - _Cino Hilliard_, Sep 25 2005

%C a(n), n>0 = term (1,1) in the n-th power of the 2x2 matrix [1,3; 2,1]. [From Gary W. Adamson, Aug 06 2010]

%C a(n) is the number of compositions of n when there are 1 type of 1 and 6 types of other natural numbers. - _Milan Janjic_, Aug 13 2010

%C Pisano period lengths: 1, 1, 1, 4, 4, 1, 24, 4, 3, 4,120, 4, 56, 24, 4, 8,288, 3, 18, 4,... - _R. J. Mathar_, Aug 10 2012

%D John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

%H Vincenzo Librandi, <a href="/A002533/b002533.txt">Table of n, a(n) for n = 0..1000</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Albert Tarn, <a href="/A001333/a001333_1.pdf">Approximations to certain square roots and the series of numbers connected therewith</a> [Annotated scanned copy]

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,5).

%F A002533(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

%F From Mario Catalani (mario.catalani(AT)unito.it), May 03 2003: (Start)

%F G.f.: (1-x)/(1-2*x-5*x^2).

%F a(n) = (1/2)*((1+sqrt(6))^n + (1-sqrt(6))^n).

%F a(n)/A083694(n) converges to sqrt(3/2).

%F a(n)/A083695(n) converges to sqrt(2/3).

%F a(n) = a(n-1) + 3*A083694(n-1).

%F a(n) = a(n-1) + 2*A083695(n-1), n>0. (End)

%F Binomial transform of expansion of cosh(sqrt(6)*x) (A000400, with interpolated zeros). E.g.f.: exp(x)*cosh(sqrt(6)*x) - Paul Barry, May 09 2003

%F From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)

%F a(2*n+1) = 2*a(n)*a(n+1) - (-5)^n.

%F a(n)^2 - 6*A002532(n)^2 = (-5)^n. (End)

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 6^k - _Paul Barry_, Jul 25 2004

%F a(n) = Sum_{k, 0<=k<=n} A098158(n,k)*6^(n-k). - _Philippe Deléham_, Dec 26 2007

%F If p[1]=1, and p[i]=6, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - _Milan Janjic_, Apr 29 2010

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013

%p A002533:=(-1+z)/(-1+2*z+5*z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t f[n_] := Simplify[((1 + Sqrt[6])^n + (1 - Sqrt[6])^n)/2]; Array[f, 28, 0] (* Or *)

%t LinearRecurrence[{2, 5}, {1, 1}, 28] (* Or *)

%t Table[ MatrixPower[{{1, 2}, {3, 1}}, n][[1, 1]], {n, 0, 25}]

%t (* _Robert G. Wilson v_, Sep 18 2013 *)

%o (Sage) [lucas_number2(n,2,-5)/2 for n in xrange(0, 21)] # _Zerinvary Lajos_, Apr 30 2009

%o (MAGMA) [(1/2)*Floor((1+Sqrt(6))^n+(1-Sqrt(6))^n): n in [0..30]]; // _Vincenzo Librandi_, Aug 15 2011

%o (PARI) a(n)=([0,1; 5,2]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, May 10 2016

%o (PARI) x='x+O('x^30); Vec((1-x)/(1-2*x-5*x^2)) \\ _G. C. Greubel_, Jan 08 2018

%o (MAGMA) [n le 2 select 1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 08 2018

%Y The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified July 20 14:19 EDT 2019. Contains 325184 sequences. (Running on oeis4.)