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A077412 Chebyshev U(n,x) polynomial evaluated at x=8. 13
1, 16, 255, 4064, 64769, 1032240, 16451071, 262184896, 4178507265, 66593931344, 1061324394239, 16914596376480, 269572217629441, 4296240885694576, 68470281953483775, 1091228270370045824, 17391182043967249409 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 16'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

For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,15}. - Milan Janjic, Jan 23 2015

LINKS

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

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (16,-1).

FORMULA

a(n) = 16*a(n-1) - a(n-2), n>=1, a(-1)=0, a(0)=1.

a(n) = S(n, 16) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1 - 16*x + x^2).

a(n) = (((8 + 3*sqrt(7))^(n+1) - (8 - 3*sqrt(7))^(n+1)))/(6*sqrt(7)).

a(n) = sqrt(A001081(n+1)^2-1)/63).

a(n) = Sum_{k=0..n} A101950(n,k)*15^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/7*(7 + 3*sqrt(7)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/16*(7 + 3*sqrt(7)). - Peter Bala, Dec 23 2012

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 8]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

CoefficientList[Series[1/(1 - 16 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)

PROG

(Sage) [lucas_number1(n, 16, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008

(MAGMA) I:=[1, 16, 255]; [n le 3 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

CROSSREFS

Sequence in context: A228982 A158531 A171321 * A208498 A207586 A208071

Adjacent sequences:  A077409 A077410 A077411 * A077413 A077414 A077415

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified November 24 07:58 EST 2017. Contains 295173 sequences.