Search: id:A054320
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%I A054320
%S A054320 1,11,109,1079,10681,105731,1046629,10360559,102558961,1015229051,
%T A054320 10049731549,99482086439,984771132841,9748229241971,96497521286869,
%U A054320 955226983626719,9455772314980321,93602496166176491,926569189346784589
%N A054320 G.f.: (1+x)/(1-10*x+x^2).
%C A054320 Chebyshev's even indexed U-polynomials evaluated at sqrt(3).
%C A054320 a(n)^2 is a star number (A003154).
%C A054320 a(n) = L(n,-10)*(-1)^n, where L is defined as in A108299; see also A072256 
               for L(n,+10). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 01 2005
%C A054320 (sqrt(2)+sqrt(3))^(2*n+1)=a(n)*sqrt(2)+A138288(n)*sqrt(3); a(n)=A138288(n)+A001078(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008
%H A054320 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A054320 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A054320 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%H A054320 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StarNumber.html">Star Number</a>
%F A054320 (a(n)-1)^2+a(n)^2+(a(n)+1)^2=b(n)^2+(b(n)+1)^2=c(n), where b(n) is A031138 
               and c(n) is A007667
%F A054320 Any k in the sequence has the successor 5*k + 2sqrt{3(2*k^2 + 1)}. - 
               Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 08 2002
%F A054320 a(n) = 10*a(n-1) - a(n-2); a(n)=(sqrt(6) - 2)/4*(5 + 2*sqrt(6))^n - (sqrt(6) 
               + 2)/4*(5 - 2*sqrt(6))^n.
%F A054320 a(n) = U(2*(n-1), sqrt(3)) = S(n-1, 10) + S(n-2, 10) with Chebyshev's 
               U(n, x) and S(n, x) := U(n, x/2) polynomials and S(-1, x) := 0. S(n, 
               10) = A004189(n+1), n>=0.
%F A054320 For all members x of the sequence, 6*x^2 + 3 is a square. Lim. n-> Inf. 
               a(n)/a(n-1) = 5 + 2*sqrt(6) - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A054320 a(n) = [ [(5+2*sqrt(6))^n - (5-2*sqrt(6))^n] + [(5+2*sqrt(6))^(n-1) - 
               (5-2*sqrt(6))^(n-1)] / (4*sqrt(6)) - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A054320 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, 
               -12)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002
%F A054320 a(n) = A001079(n) + 3*A001078(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 12 2008
%F A054320 A054320(n) = A142238(2n) = A041006(2n)/2 = A041038(2n)/4 [From M. F. 
               Hasler (MHasler(AT)univ-ag.fr), Feb 14 2009]
%e A054320 a(1)^2=121 is the 5th star number (A003154).
%t A054320 q=12;s=0;lst={};Do[s+=n;If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]],AppendTo[lst,
               Sqrt[q*s+1]]],{n,0,9!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Apr 02 2009]
%o A054320 (PARI) a(n)=if(n<1,0,subst(poltchebi(n)-poltchebi(n-1),x,5)/4)
%o A054320 (Other) sage: [(lucas_number2(n,10,1)-lucas_number2(n-1,10,1))/8 for 
               n in xrange(1, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 10 2009]
%Y A054320 A member of the family A057078, A057077, A057079, A005408, A002878, A001834, 
               A030221, A002315, A033890, A057080, A057081, A054320, which are the 
               expansions of (1+x) / (1-kx+x^2) with k = -1, 0, 1, 2, 3, 4, 5, 6, 
               7, 8, 9, 10 . Philippe DELEHAM, May 04 2004
%Y A054320 Cf. A003154, A031138, A007667, A004189. a(n) = sqrt((3* A072256(n)^2 
               - 1)/2).
%Y A054320 Cf. A138281.
%Y A054320 Sequence in context: A125423 A165149 A048346 this_sequence A124290 A094703 
               A144744
%Y A054320 Adjacent sequences: A054317 A054318 A054319 this_sequence A054321 A054322 
               A054323
%K A054320 easy,nonn,new
%O A054320 0,2
%A A054320 Ignacio Larrosa Canestro (ignacio.larrosa(AT)eresmas.net) Feb 27 2000
%E A054320 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Oct 31 2002

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