A077421 Chebyshev sequence U(n,11)=S(n,22) with Diophantine property.

1, 22, 483, 10604, 232805, 5111106, 112211527, 2463542488, 54085723209, 1187422368110, 26069206375211, 572335117886532, 12565303387128493, 275864339398940314, 6056450163389558415, 132966039255171344816

Offset: 0

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

Comments

b(n)^2 - 30*(2*a(n))^2 = 1 with the companion sequence b(n)=A077422(n+1).

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 22's along the main diagonal, and i's along the subdiagonal and the superdiagonal (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,...,21}. - Milan Janjic, Jan 25 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..700
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (22,-1).
• Index entries for sequences related to Chebyshev polynomials.

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), A077412 (m=8), A049660 (m=9), A075843 (m=10), this sequence (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:=11;; 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 23 2019

Magma

I:=[1, 22]; [n le 2 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

Maple

seq( simplify(ChebyshevU(n, 11)), n=0..20); # G. C. Greubel, Dec 23 2019

Mathematica

Table[GegenbauerC[n, 1, 11], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[1/(1-22x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
ChebyshevU[Range[21] -1, 11] (* G. C. Greubel, Dec 23 2019 *)

PARI

vector( 21, n, polchebyshev(n-1, 2, 11) ) \\ G. C. Greubel, Dec 23 2019

Sage

[lucas_number1(n,22,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Sage

[chebyshev_U(n,11) for n in (0..20)] # G. C. Greubel, Dec 23 2019

Formula

a(n) = 22*a(n-1) - a(n-1), a(-1)=0, a(0)=1.

a(n) = S(n, 22) 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 := 11+2*sqrt(30) and am := 11-2*sqrt(30).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*22^(n-2*k).

a(n) = sqrt((A077422(n+1)^2-1)/30)/2.

G.f.: 1/(1-22*x+x^2). - Philippe Deléham, Nov 18 2008

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*21^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(30)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/11*(5 + sqrt(30)). - Peter Bala, Dec 23 2012