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%I #58 Sep 08 2022 08:45:07
%S 1,22,483,10604,232805,5111106,112211527,2463542488,54085723209,
%T 1187422368110,26069206375211,572335117886532,12565303387128493,
%U 275864339398940314,6056450163389558415,132966039255171344816
%N Chebyshev sequence U(n,11)=S(n,22) with Diophantine property.
%C b(n)^2 - 30*(2*a(n))^2 = 1 with the companion sequence b(n)=A077422(n+1).
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A077421/b077421.txt">Table of n, a(n) for n = 0..700</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = 22*a(n-1) - a(n-1), a(-1)=0, a(0)=1.
%F a(n) = S(n, 22) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310.
%F a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap := 11+2*sqrt(30) and am := 11-2*sqrt(30).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*22^(n-2*k).
%F a(n) = sqrt((A077422(n+1)^2-1)/30)/2.
%F G.f.: 1/(1-22*x+x^2). - _Philippe Deléham_, Nov 18 2008
%F a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*21^k. - _Philippe Deléham_, Feb 10 2012
%F Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(30)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 1} (1 - 1/a(n)) = 1/11*(5 + sqrt(30)). - _Peter Bala_, Dec 23 2012
%p seq( simplify(ChebyshevU(n, 11)), n=0..20); # _G. C. Greubel_, Dec 23 2019
%t Table[GegenbauerC[n, 1, 11], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t CoefficientList[Series[1/(1-22x+x^2), {x,0,20}], x] (* _Vincenzo Librandi_, Dec 24 2012 *)
%t ChebyshevU[Range[21] -1, 11] (* _G. C. Greubel_, Dec 23 2019 *)
%o (Sage) [lucas_number1(n,22,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o (Sage) [chebyshev_U(n,11) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o (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
%o (PARI) vector( 21, n, polchebyshev(n-1, 2, 11) ) \\ _G. C. Greubel_, Dec 23 2019
%o (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
%Y 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).
%Y Cf. A323182.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 29 2002
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