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A052918 a(0)=1, a(1)=5, a(n+1) = 5*a(n) + a(n-1). 29
1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, 51872282415, 269351100901, 1398627786920, 7262490035501, 37711077964425, 195817879857626 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

[A085448(n)]^2 - 29*[a(n-1)]^2 = 4*(-1)^n. - Gary W. Adamson, Jul 01 2003, corrected Oct 07 2008

a(p) == 29^((p-1)/2)) mod p, for odd primes p. - Gary W. Adamson, Feb 22 2009

For more information about this type of recurrence, follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

Binomial transform of A015523. - Johannes W. Meijer, Aug 01 2010

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 5's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

a(n) equals the number of words of length n on alphabet {0,1,...,5} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 901

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,1).

FORMULA

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

a(3n) = A041047(5n), a(3n+1) = A041047(5n+3), a(3n+2) = 2*A041047(5n+4). - Henry Bottomley, May 10 2000

Sum(1/29*(5+2*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+5*_Z+_Z^2)).

a(n-1) = (((5 + sqrt(29))/2)^n - ((5 - sqrt(29))/2)^n)/sqrt(29). - Gary W. Adamson, Jul 01 2003

a(n)= U(n, 5*I/2)*(-I)^n with I^2=-1 and Chebyshev's U(n, x/2)=S(n, x) polynomials. See triangle A049310.

Let M = {{0, 1}, {1, 5}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a = v[n][[1]]. - Roger L. Bagula, May 29 2005

a(n)=F(n, 5), the n-th Fibonacci polynomial evaluated at x=5. - T. D. Noe, Jan 19 2006

a(n), n>0 = denominator of n-th convergent to [1, 4, 5, 5, 5,...]. Continued fraction [1, 4, 5, 5, 5,...] = .807417596..., the inradius of a right triangle with legs 2 and 5. n-th convergent = A100237(n)/A052918(n), the first few being: 1/1, 4/5, 21/26, 109/135, 566/701,... - Gary W. Adamson, Dec 21 2007

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2n+1) = 5*A097781(n), a(2n) = A097835(n).

Limit(a(n+k)/a(k), k=infinity) = (A087130(n) + A052918(n-1)*sqrt(29))/2.

Limit(A087130(n)/A052918(n-1), n=infinity) = sqrt(29).

(End)

Limit(a(n+k)/a(k),k=infinity) = (A087130(n)+A052918(n-1)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010

From L. Edson Jeffery, Jan 07 2012: (Start)

Define the 2 X 2 matrix A=[1,1; 5,4]. Then:

a(n) = (1/5)*[A^(n+2)-A^(n+1)]_(1,1);

a(n) = [A^(n+1)]_(1,2);

a(n) = (1/5)*[A^(n+1)]_(2,1);

a(n) = [sum[k=0..n, A^k]]_(2,2). (End)

sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = (sqrt(29)-5)/2. - Vladimir Shevelev, Feb 23 2013

MAPLE

spec := [S, {S=Sequence(Union(Z, Z, Z, Z, Z, Prod(Z, Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

a[0]:=1: a[1]:=5: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006

with(combinat):a:=n->fibonacci(n, 5):seq(a(n), n=1..22); # Zerinvary Lajos, Dec 07 2008

MATHEMATICA

  a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*5, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

LinearRecurrence[{5, 1}, {1, 5}, 30] (* Vincenzo Librandi, Feb 23 2013 *)

Table[Fibonacci[n + 1, 5], {n, 0, 20}] (* Vladimir Reshetnikov, May 08 2016 *)

PROG

(Sage) [lucas_number1(n, 5, -1) for n in xrange(1, 22)] # Zerinvary Lajos, Apr 24 2009

(PARI) Vec(-1/(-1+5*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011

(MAGMA) I:=[1, 5]; [n le 2 select I[n] else 5*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 23 2013

CROSSREFS

Cf. A000045, A000129, A006190, A001076, A005668, A085448, A099365 (squares), A100237, A175184 (Pisano periods), A201005 (prime subsequence).

Cf. A243399.

Sequence in context: A047768 A022032 A255118 * A255633 A255815 A018903

Adjacent sequences:  A052915 A052916 A052917 * A052919 A052920 A052921

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

STATUS

approved

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Last modified June 24 20:03 EDT 2017. Contains 288707 sequences.