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A056771
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a(n) = a(-n) = 34*a(n-1) - a(n-2), and a(0)=1, a(1)=17.
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8
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1, 17, 577, 19601, 665857, 22619537, 768398401, 26102926097, 886731088897, 30122754096401, 1023286908188737, 34761632124320657, 1180872205318713601, 40114893348711941777, 1362725501650887306817, 46292552162781456490001
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The sequence satisfies the Pell equation a(n)^2 - 18 * A202299(n+1)^2 = 1. - Vincenzo Librandi, Dec 19 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 to sequences with linear recurrences with constant coefficients, signature (34,-1).
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FORMULA
| a(n) = (r^n+1/r^n)/2 with r = 17+sqrt(17^2-1).
a(n) = 16*A001110(n)+1 = A001541(2n) = (4*A001109(n))^2+1 = 3*A001109(2n-1)-A001109(2n-2) = A001109(2n)-3*A001109(2n-1).
a(n) = T(n, 17) = T(2*n, 3) with T(n, x) Chebyshev's polynomials of the first kind. See A053120. T(n, 3)= A001541(n).
G.f.: x*(1-17*x)/(1-34*x+x^2).
a(n) = Cosh(2n*ArcSinh(Sqrt(8))) - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008
a(n) = (a^n + b^n)/2 where a=17+12sqrt(2) and b=17-12sqrt(2). Sqrt(a(n)-1)/4 = A001109(n). - James R. Buddenhagen, Dec 09 2011
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MATHEMATICA
| LinearRecurrence[{34, -1}, {1, 17}, 30] (* Vincenzo Librandi, Dec 18 2011 *)
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PROG
| sage: [lucas_number2(n, 34, 1)/2 for n in xrange(0, 15)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
(MAGMA) I:=[1, 17]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 18 2011
(Maxima) makelist(expand(((17+sqrt(288))^n+(17-sqrt(288))^n))/2, n, 0, 15); // Vincenzo Librandi, Dec 18 2011
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CROSSREFS
| Cf. A001075, A001541, A001091, A001079, A023038, A011943, A001081, A023039, A001085 and note relationship with square triangular number sequences A001110 and A001109.
Row 3 of array A188644
Sequence in context: A197395 A012069 A191865 * A041547 A041544 A202407
Adjacent sequences: A056768 A056769 A056770 * A056772 A056773 A056774
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KEYWORD
| nonn,easy
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Aug 16 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 07 2000
Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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