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A070998 a(n) = 9*a(n-1) - a(n-2) for n>0, a(0)=1, a(-1)=1. 12
1, 8, 71, 631, 5608, 49841, 442961, 3936808, 34988311, 310957991, 2763633608, 24561744481, 218292066721, 1940066856008, 17242309637351, 153240719880151, 1361924169284008, 12104076803675921, 107574767063799281, 956068826770517608 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A Pellian sequence.

In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1); - Paul Barry, Mar 13 2005

a(n) = L(n,9), where L is defined as in A108299; see also A057081 for L(n,-9). - Reinhard Zumkeller, Jun 01 2005

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

For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(7)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 08 2011]

Positive values of x (or y) satisfying x^2 - 9xy + y^2 + 7 = 0. - Colin Barker, Feb 09 2014

LINKS

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

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, 2014

Index entries for linear recurrences with constant coefficients, signature (9,-1).

FORMULA

a(n) ~ 1/11*sqrt(11)*(1/2*(sqrt(11)+sqrt(7)))^(2*n+1)

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 7)=a(n) - Benoit Cloitre, Nov 10 2002

a(n)*a(n+3) = 63 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004

a(n)=(-1)^n*U(2n, I*sqrt(7)/2), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry, Mar 13 2005

G.f.: (1-x)/(1-9*x+x^2). [Philippe Deléham, Nov 03 2008]

a(n) = (1/2)*[(9/2)+(1/2)*sqrt(77)]^(n+1)+(1/22)*[(9/2)-(1/2)*sqrt(77)]^(n+1)*sqrt(77)-(1/22)*[(9/2)+(1/2) *sqrt(77)]^(n+1)*sqrt(77)+(1/2)*[(9/2)-(1/2)*sqrt(77)]^(n+1). [Paolo P. Lava, Nov 20 2008]

a(n) = A018913(n+1)-A018913(n). - R. J. Mathar, Jun 07 2013

MATHEMATICA

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

LinearRecurrence[{9, -1}, {1, 8}, 30] (* Harvey P. Dale, Sep 24 2015 *)

PROG

(Sage) [lucas_number1(n, 9, 1)-lucas_number1(n-1, 9, 1) for n in xrange(1, 19)]# [From Zerinvary Lajos, Nov 10 2009]

(MAGMA) I:=[1, 8]; [n le 2 select I[n] else 9*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 10 2014

CROSSREFS

Cf. A057081, A056918.

Row 9 of array A094954.

Cf. similar sequences listed in A238379.

Sequence in context: A038145 A198856 A015576 * A187709 A152265 A081178

Adjacent sequences:  A070995 A070996 A070997 * A070999 A071000 A071001

KEYWORD

nonn,easy

AUTHOR

Joe Keane (jgk(AT)jgk.org), May 18 2002

EXTENSIONS

More terms from Vincenzo Librandi, Feb 10 2014

STATUS

approved

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Last modified May 25 01:01 EDT 2017. Contains 287008 sequences.