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A072256 a(n) = 10*a(n-1) - a(n-2) for n>1, a(0) = a(1) = 1. 25
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).

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

REFERENCES

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

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.

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

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Last modified March 25 21:38 EDT 2017. Contains 284111 sequences.