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A072256
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a(n) = 10*a(n-1) - a(n-2) for n>1, a(0) = a(1) = 1.
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25
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1, 1, 9, 89, 881, 8721, 86329, 854569, 8459361, 83739041, 828931049, 8205571449, 81226783441, 804062262961, 7959395846169, 78789896198729, 779939566141121, 7720605765212481, 76426118085983689, 756540575094624409
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 283).
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LINKS
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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)
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Lekraj Beedassy, Jul 08 2002
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EXTENSIONS
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Edited by Robert G. Wilson v, Jul 17 2002
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STATUS
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approved
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