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A097781
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Chebyshev polynomials S(n,27) with Diophantine property.
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6
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1, 27, 728, 19629, 529255, 14270256, 384767657, 10374456483, 279725557384, 7542215592885, 203360095450511, 5483180361570912, 147842509666964113, 3986264580646460139, 107481301167787459640, 2898008866949614950141
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not squarefree.
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 27's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences relate d to Chebyshev polynomials.
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FORMULA
| a(n)= S(n, 27)=U(n, 27/2)= S(2*n+1, sqrt(29))/sqrt(29) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n)=27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0.
a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (27+5*sqrt(29))/2 and am := (27-5*sqrt(29))/2.
G.f.: 1/(1-27*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*26^k. - DELEHAM Philippe, Feb 10 2012
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EXAMPLE
| (x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4.
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MAPLE
| with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008
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MATHEMATICA
| Join[{a=1, b=27}, Table[c=27*b-a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)
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PROG
| sage: [lucas_number1(n, 27, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| Cf. A078362, A078366.
Sequence in context: A170708 A170746 A171332 * A073537 A016947 A167726
Adjacent sequences: A097778 A097779 A097780 * A097782 A097783 A097784
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
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