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A077421 Chebyshev sequence U(n,11)=S(n,22) with Diophantine property. 8
1, 22, 483, 10604, 232805, 5111106, 112211527, 2463542488, 54085723209, 1187422368110, 26069206375211, 572335117886532, 12565303387128493, 275864339398940314, 6056450163389558415, 132966039255171344816 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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

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

MATHEMATICA

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

CoefficientList[Series[1/(1 - 22 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)

PROG

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

(MAGMA) I:=[1, 22, 483]; [n le 3 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

CROSSREFS

Sequence in context: A261135 A158535 A171327 * A207491 A207888 A208111

Adjacent sequences:  A077418 A077419 A077420 * A077422 A077423 A077424

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified November 20 04:05 EST 2017. Contains 294959 sequences.