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A029547
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Expansion of 1/(1-34*x+x^2).
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10
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1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Chebyshev sequence U(n,17)=S(n,34) with Diophantine property.
b(n)^2 - 2*(12*a(n))^2 = 1 with the companion sequence b(n)=A056771(n+1). - Wolfdieter Lang, Dec 11 2002
More generally, for t(m)=m+sqrt(m^2-1) and u(n)=(t(m)^(n+1)-1/t(m)^(n+1))/(t(m)-1/t(m)), we can verify that ((u(n+1)-u(n-1))/2)^2-(m^2-1)*u(n)^2=1. - Bruno Berselli, Nov 21 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (34,-1).
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FORMULA
| a(n) = 34*a(n-1) - a(n-2), a(-1)=0, a(0)=1.
a(n) = S(n, 34) with S(n, x):= U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Dec 11 2002
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = 17+12*sqrt(2) and am = 17-12*sqrt(2).
a(n) = sum((-1)^k*binomial(n-k, k)*34^(n-2*k), k = 0..floor(n/2)).
a(n) = sqrt((A056771(n+1)^2 -1)/2)/12.
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) with a(-1)=0, a(0)=1, a(1)=34. Also a(n) = (sqrt(2)/48)*((17+12*sqrt(2))^n-(17-12*sqrt(2))^n) = (sqrt(2)/48)*((3+2*sqrt(2))^(2n+2)-(3-2*sqrt(2))^(2n+2)) = (sqrt(2)/48)*((1+sqrt(2))^(4n+4)-(1-sqrt(2))^(4n+4)). - Antonio A. Olivares, Mar 19 2008
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*33^k. - DELEHAM Philippe, Feb 10 2012
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MAPLE
| with (combinat):seq(fibonacci(4*n, 2)/12, n=1..15); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2008
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 17]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky, Sep 11 2008]
LinearRecurrence[{34, -1}, {1, 34}, 20] (* Vincenzo Librandi, Nov 22 2011 *)
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PROG
| (PARI) A029547(n, x=[0, 1], A=[17, 72*4; 1, 17]) = vector(n, i, (x*=A)[1]) - M. F. Hasler, May 26 2007
(Other) sage: [lucas_number1(n, 34, 1) for n in xrange(1, 16)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]
(MAGMA) I:=[1, 34]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011
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CROSSREFS
| A091761 is an essentially identical sequence.
Cf. A200441, A200724.
Sequence in context: A170753 A158696 * A091761 A009978 A041545 A189434
Adjacent sequences: A029544 A029545 A029546 * A029548 A029549 A029550
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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