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A070998 a(n) = 9*a(n-1) - a(n-2) for n > 0, a(0)=1, a(-1)=1. 13
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. - 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

REFERENCES

Fink, Alex, Richard Guy, and Mark Krusemeyer. "Partitions with parts occurring at most thrice." Contributions to Discrete Mathematics 3.2 (2008), 76-114. See Section 13.

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 [math.CO], 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)] # 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: A198856 A320092 A015576 * A187709 A292865 A152265

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 October 16 19:22 EDT 2018. Contains 316271 sequences. (Running on oeis4.)