A077412 Chebyshev U(n,x) polynomial evaluated at x=8.

1, 16, 255, 4064, 64769, 1032240, 16451071, 262184896, 4178507265, 66593931344, 1061324394239, 16914596376480, 269572217629441, 4296240885694576, 68470281953483775, 1091228270370045824, 17391182043967249409

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

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
• Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (16,-1).

Crossrefs

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), this sequence (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).

Cf. A323182.

Programs

GAP

m:=8;; a:=[1,2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

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

Maple

seq( simplify(ChebyshevU(n, 8)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

Table[GegenbauerC[n, 1, 8], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[1/(1-16x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
LinearRecurrence[{16,-1}, {1,16}, 30] (* G. C. Greubel, Jan 18 2018 *)
ChebyshevU[Range[21] -1, 8] (* G. C. Greubel, Dec 22 2019 *)

PARI

vector( 21, n, polchebyshev(n-1, 2, 8) ) \\ G. C. Greubel, Jan 18 2018

Sage

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

Sage

[chebyshev_U(n,8) for n in (0..20)] # G. C. Greubel, Dec 22 2019

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