Search: id:A004191
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%I A004191
%S A004191 1,12,143,1704,20305,241956,2883167,34356048,409389409,
%T A004191 4878316860,58130412911,692686638072,8254109243953,98356624289364,
%U A004191 1172025382228415,13965947962451616,166419350167190977
%N A004191 Expansion of 1/(1-12*x+x^2).
%C A004191 Chebyshev's polynomials U(n,x) evaluated at x=6.
%C A004191 a(n) give all (nontrivial, integer) solutions of Pell equation b(n)^2 
               - 35*a(n)^2 = +1 with b(n)=A023038(n+1), n>=0.
%H A004191 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A004191 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A004191 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A004191 a(n) = S(n, 12) with S(n, x) := U(n, x/2) Chebyshev's polynomials of 
               the second kind. See A049310.
%F A004191 a(n) = ((6+sqrt(35))^(n+1) - (6-sqrt(35))^(n+1))/(2*sqrt(35)).
%F A004191 a(n) = sqrt((A023038(n)^2 - 1)/35).
%F A004191 [A077417(n), a(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2008
%F A004191 a(n)=12*a(n-1)-a(n-2)for n>1, a(0)=1, a(1)=12. [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 17 2008]
%t A004191 lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 6]], {n, 0, 8^2}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
%o A004191 sage: [lucas_number1(n,12,1) for n in xrange(1,20)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%Y A004191 Cf. A077417.
%Y A004191 Sequence in context: A056330 A158516 A163448 this_sequence A051051 A001021 
               A159490
%Y A004191 Adjacent sequences: A004188 A004189 A004190 this_sequence A004192 A004193 
               A004194
%K A004191 nonn
%O A004191 0,2
%A A004191 N. J. A. Sloane (njas(AT)research.att.com).
%E A004191 Chebyshev comments and a(n) formulas from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 08 2002

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