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A078366 A Chebyshev S-sequence with Diophantine property. 7
1, 17, 288, 4879, 82655, 1400256, 23721697, 401868593, 6808044384, 115334885935, 1953885016511, 33100710394752, 560758191694273, 9499788548407889, 160935647131239840, 2726406212682669391 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 285*a^2 =+4 with companion sequence b(n)=A078367(n+1), n>=0.

This is the m=19 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..18 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364 and A077412. The m=1..3 (signed) sequences are A049347, A056594, A010892.

For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 17's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

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=17, 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=19.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index to sequences with linear recurrences with constant coefficients, signature (17,-1).

FORMULA

a(n)=17*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.

a(n)=S(2*n+1, sqrt(19))/sqrt(19) = S(n, 17); 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 := (17+sqrt(285))/2 and am := (17-sqrt(285))/2.

G.f.: 1/(1-17*x+x^2).

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*16^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/15*(15 + sqrt(285)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/34*(15 + sqrt(285)). - Peter Bala, Dec 23 2012

MATHEMATICA

Join[{a=1, b=17}, Table[c=17*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011* )

CoefficientList[Series[1/(1 - 17 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)

PROG

(Sage) [lucas_number1(n, 17, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008

(MAGMA) I:=[1, 17, 288]; [n le 3 select I[n] else 17*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

CROSSREFS

a(n)=sqrt((A078367(n+1)^2 - 4)/285), n>=0, (Pell equation d=285, +4).

Cf. A077428, A078355 (Pell +4 equations).

Sequence in context: A196743 A196901 A171322 * A045607 A045606 A171291

Adjacent sequences:  A078363 A078364 A078365 * A078367 A078368 A078369

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified December 21 20:22 EST 2014. Contains 252326 sequences.