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 A002533 a(n) = 2*a(n-1) + 5*a(n-2). (Formerly M4369 N1834) 29
 1, 1, 7, 19, 73, 241, 847, 2899, 10033, 34561, 119287, 411379, 1419193, 4895281, 16886527, 58249459, 200931553, 693110401, 2390878567, 8247309139, 28449011113, 98134567921, 338514191407, 1167701222419, 4027973401873 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 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] 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 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 REFERENCES John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy] Index entries for linear recurrences with constant coefficients, signature (2,5). FORMULA A002533(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003 From Mario Catalani (mario.catalani(AT)unito.it), May 03 2003: (Start) G.f.: (1-x)/(1-2*x-5*x^2). a(n) = (1/2)*((1+sqrt(6))^n + (1-sqrt(6))^n). a(n)/A083694(n) converges to sqrt(3/2). a(n)/A083695(n) converges to sqrt(2/3). a(n) = a(n-1) + 3*A083694(n-1). a(n) = a(n-1) + 2*A083695(n-1), n>0. (End) 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 From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start) a(2*n+1) = 2*a(n)*a(n+1) - (-5)^n. a(n)^2 - 6*A002532(n)^2 = (-5)^n. (End) a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 6^k - Paul Barry, Jul 25 2004 a(n) = Sum_{k, 0<=k<=n} A098158(n,k)*6^(n-k). - Philippe Deléham, Dec 26 2007 If p=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 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 MAPLE A002533:=(-1+z)/(-1+2*z+5*z**2); # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA f[n_] := Simplify[((1 + Sqrt)^n + (1 - Sqrt)^n)/2]; Array[f, 28, 0] (* Or *) LinearRecurrence[{2, 5}, {1, 1}, 28] (* Or *) Table[ MatrixPower[{{1, 2}, {3, 1}}, n][[1, 1]], {n, 0, 25}] (* Robert G. Wilson v, Sep 18 2013 *) PROG (Sage) [lucas_number2(n, 2, -5)/2 for n in range(0, 21)] # Zerinvary Lajos, Apr 30 2009 (MAGMA) [(1/2)*Floor((1+Sqrt(6))^n+(1-Sqrt(6))^n): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011 (PARI) a(n)=([0, 1; 5, 2]^n*[1; 1])[1, 1] \\ Charles R Greathouse IV, May 10 2016 (PARI) x='x+O('x^30); Vec((1-x)/(1-2*x-5*x^2)) \\ G. C. Greubel, Jan 08 2018 (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 CROSSREFS The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519. Sequence in context: A318483 A005516 A152008 * A111011 A144723 A062551 Adjacent sequences:  A002530 A002531 A002532 * A002534 A002535 A002536 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified September 18 20:19 EDT 2020. Contains 337173 sequences. (Running on oeis4.)