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A070998
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a(n) = 9*a(n-1) - a(n-2), a(0)=1, a(-1)=1.
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7
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1, 8, 71, 631, 5608, 49841, 442961, 3936808, 34988311, 310957991, 2763633608, 24561744481, 218292066721, 1940066856008, 17242309637351, 153240719880151, 1361924169284008, 12104076803675921
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OFFSET
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0,2
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COMMENTS
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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]
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LINKS
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Table of n, a(n) for n=0..17.
Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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FORMULA
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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). - R. 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). [From Philippe DELEHAM, 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), with n>=0 [From Paolo P. Lava, Nov 20 2008]
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PROG
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(Sage) [lucas_number1(n, 9, 1)-lucas_number1(n-1, 9, 1) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009]
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CROSSREFS
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Cf. A057081, A056918.
Row 9 of array A094954.
Sequence in context: A038145 A198856 A015576 * A187709 A152265 A081178
Adjacent sequences: A070995 A070996 A070997 * A070999 A071000 A071001
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KEYWORD
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nonn
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AUTHOR
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Joe Keane (jgk(AT)jgk.org), May 18 2002
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STATUS
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approved
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