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A077423
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Chebyshev sequence U(n,12)=S(n,24) with Diophantine property.
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1
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1, 24, 575, 13776, 330049, 7907400, 189447551, 4538833824, 108742564225, 2605282707576, 62418042417599, 1495427735314800, 35827847605137601, 858372914787987624, 20565122107306565375, 492704557660569581376
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| b(n)^2 - 143*a(n)^2 = 1 with the companion sequence b(n)=A077424(n+1).
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 24's along the main diagonal, and i's along the subdiagonal and the superdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
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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)=24*a(n-1) - a(n-2), a(-1) := 0, a(0)=1.
a(n)= S(n, 24) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310.
a(n)= (ap^(n+1) - am^(n+1))/(ap - am) with ap := 12+sqrt(143) and am := 12-sqrt(143).
a(n)= sum(((-1)^k)*binomial(n-k, k)*24^(n-2*k), k=0..floor(n/2)).
a(n)=sqrt((A077424(n+1)^2 - 1)/143).
G.f.: 1/(1-24*x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*23^k. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 12]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
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PROG
| sage: [lucas_number1(n, 24, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| Sequence in context: A007109 A158538 A171329 * A059061 A206991 A206933
Adjacent sequences: A077420 A077421 A077422 * A077424 A077425 A077426
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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