login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A072256 a(n) = 10*a(n-1) - a(n-2) for n > 1, a(0) = a(1) = 1. 27
1, 1, 9, 89, 881, 8721, 86329, 854569, 8459361, 83739041, 828931049, 8205571449, 81226783441, 804062262961, 7959395846169, 78789896198729, 779939566141121, 7720605765212481, 76426118085983689, 756540575094624409 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Any k in the sequence is followed by 5*k + 2sqrt{2(3*k^2 - 1)}.

Gives solutions for x in 3*x^2 - 2*y^2 = 1. Corresponding y is given by A054320(n-1). [corrected by Jon E. Schoenfield, Jun 08 2018]

a(n) = L(n-1,10), where L is defined as in A108299; see also A054320 for L(n,-10). - Reinhard Zumkeller, Jun 01 2005

Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8,9} which do not end in 0. - Tanya Khovanova, Jan 10 2007

a(n) = A138288(n-1) for n > 0. - Reinhard Zumkeller, Mar 12 2008

For n>= 2, a(n) equals the permanent of the (2n-2)X(2n-2) tridiagonal matrix with sqrt(8)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 8 = 0. - Colin Barker, Feb 09 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Bruno Deschamps, Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 283).

Tanya Khovanova, Recursive Sequences

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Index entries for sequences related to Chebyshev polynomials

Index entries for linear recurrences with constant coefficients, signature (10,-1)

FORMULA

a(n) = (3-sqrt(6))/6 * (5+2*sqrt(6))^n + (3+sqrt(6))/6 * (5-2*sqrt(6))^n.

a(n) = {2*A031138(n) + 1}/3 = sqrt(2*A054320(n)^2 + 1)/3), n>=1.

a(n) = U(n-1, 5)-U(n-2, 5) = T(2*n-1, sqrt(3))/sqrt(3) with Chebyshev's U- and T- polynomials and U(-1, x) := 0, U(-2, x) := -1, T(-1, x) := x.

G.f.: (1-9*x)/(1-10*x+x^2).

For all members x of the sequence, 6*x^2 - 2 is a square. Lim. n -> Inf. a(n)/a(n-1) = 5 + 2*sqrt(6). - Gregory V. Richardson, Oct 10 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 8)=a(n+1). - Benoit Cloitre, Nov 10 2002

a(n)*a(n+3) = 80 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004

a(n) = sqrt(A046172(n)). - Paul Weisenhorn, May 15 2009

MATHEMATICA

a[n_] := a[n] = 10a[n - 1] - a[n - 2]; a[0] = a[1] = 1; Table[ a[n], {n, 0, 20}]

CoefficientList[Series[(1 - 9 x)/(1 - 10 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)

PROG

(Sage) [lucas_number1(n, 10, 1)-lucas_number1(n-1, 10, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Nov 10 2009

(MAGMA) [n le 2 select 1 else 10*Self(n-1)-Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 10 2014

(PARI) a(n)=([0, 1; -1, 10]^n*[1; 1])[1, 1] \\ Charles R Greathouse IV, May 10 2016

CROSSREFS

Cf. A054320, A031138, A046172.

Row 10 of array A094954.

First differences of A004189.

Sequence in context: A198967 A015584 * A138288 A059482 A109002 A142991

Adjacent sequences:  A072253 A072254 A072255 * A072257 A072258 A072259

KEYWORD

nonn,easy

AUTHOR

Lekraj Beedassy, Jul 08 2002

EXTENSIONS

Edited by Robert G. Wilson v, Jul 17 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 20 17:50 EDT 2018. Contains 312817 sequences. (Running on oeis4.)