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A023038
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a(n) = 12a(n-1) - a(n-2).
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9
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1, 6, 71, 846, 10081, 120126, 1431431, 17057046, 203253121, 2421980406, 28860511751, 343904160606, 4097989415521, 48831968825646, 581885636492231, 6933795669081126, 82623662392481281, 984550153040694246
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Chebyshev's polynomials T(n,x) evaluated at x=6.
a(n+1) give all (nontrivial, integer) solutions of Pell equation a(n+1)^2 - 35*b(n)^2 = +1 with b(n)=A004191(n), n>=0.
a(35+70k)-1 and a(35+70k)+1 are consecutive odd powerful numbers. The first pair is 23101441813552306872262673994181386126+-1. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006
Numbers n such that 35*(n^2-1) is a square. [From Vincenzo Librandi, Nov 19 2010]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n) = T(n, 6) = (S(n, 12)-S(n-2, 12))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 12)=A004191(n).
a(n) = ((6+sqrt(35))^n + (6-sqrt(35))^n)/2.
G.f.: (1-6*x)/(1-12*x+x^2).
a(n)a(n+3) - a(n+1)a(n+2) = 420. - R. Stephan, Jun 06 2005
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CROSSREFS
| Cf. A087800.
Sequence in context: A050788 A027317 A099339 * A092660 A186658 A092085
Adjacent sequences: A023035 A023036 A023037 * A023039 A023040 A023041
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
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