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A078364
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A Chebyshev S-sequence with Diophantine property.
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7
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1, 15, 224, 3345, 49951, 745920, 11138849, 166336815, 2483913376, 37092363825, 553901543999, 8271430796160, 123517560398401, 1844491975179855, 27543862067299424, 411313439034311505
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) gives the general (positive integer) solution of the Pell equation b^2 - 221*a^2 =+4 with companion sequence b(n)=A078365(n+1), n>=0.
This is the m=17 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..16 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362 and A007655. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 15'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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REFERENCES
| A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=15, q=-1.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=17.
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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.
Harvey P. Dale, Table of n, a(n) for n = 0..850
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FORMULA
| a(n)=15*a(n-1)-a(n-2), n>= 1; a(-1)=0, a(0)=1.
a(n)=S(2*n+1, sqrt(17))/sqrt(17) = S(n, 15); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n)=(ap^(n+1)-am^(n+1))/(ap-am) with ap := (15+sqrt(221))/2 and am := (15-sqrt(221))/2.
G.f.: 1/(1-15*x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*14^k. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| LinearRecurrence[{15, -1}, {1, 15}, 30] (* From Harvey P. Dale, Oct 16 2011 *)
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PROG
| sage: [lucas_number1(n, 15, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| a(n)=sqrt((A078365(n+1)^2 - 4)/221), n>=0, (Pell equation d=221, +4).
Cf. A077428, A078355 (Pell +4 equations).
Sequence in context: A057500 A137916 A171320 * A207423 A012852 A171289
Adjacent sequences: A078361 A078362 A078363 * A078365 A078366 A078367
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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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