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Search: id:A097310
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| A097310 |
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Chebyshev T-polynomials T(n,14) with Diophantine property. |
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+0
4
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| 1, 14, 391, 10934, 305761, 8550374, 239104711, 6686381534, 186979578241, 5228741809214, 146217791079751, 4088869408423814, 114342125644787041, 3197490648645613334, 89415396036432386311, 2500433598371461203374
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n)^2 - 195 b(n)^2 = +1 with b(n):=A097311(n) gives all nonnegative solutions of this Pell equation.
a(195+390k)-1 and a(195+390k)+1 are consecutive odd powerful numbers. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006
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LINKS
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Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n)=28*a(n-1) - a(n-2), a(-1):= 14, a(0)=1.
a(n)= T(n, 14)= (S(n, 28)-S(n-2, 28))/2 = S(n, 28)-14*S(n-1, 28) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 28)=A097311(n).
a(n)= (ap^n + am^n)/2 with ap := 14+sqrt(195) and am := 14-sqrt(195).
a(n)= sum(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*14)^(n-2*k), k=0..floor(n/2)), n>=1.
G.f.: (1-14*x)/(1-28*x+x^2).
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PROGRAM
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sage: [lucas_number2(n, 28, 1)/2 for n in xrange(0, 16)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
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CROSSREFS
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Cf. a(n)=sqrt(1 + 195*A097311(n)^2), n>=0.
Cf. A090249.
Sequence in context: A113673 A159535 A000473 this_sequence A041367 A041364 A033815
Adjacent sequences: A097307 A097308 A097309 this_sequence A097311 A097312 A097313
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
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