Search: id:A054491
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%I A054491
%S A054491 1,6,23,86,321,1198,4471,16686,62273,232406,867351,3236998,12080641,
%T A054491 45085566,168261623,627960926,2343582081,8746367398,32641887511,
%U A054491 121821182646,454642843073,1696750189646,6332357915511,23632681472398
%N A054491 a(n)=4a(n-1)-a(n-2), a(0)=1, a(1)=6.
%C A054491 Bisection (even part) of Chebyshev sequence with Diophantine property.
%C A054491 The odd part is A077234(n) with Diophantine companion A077235(n).
%C A054491 -3*a(n)^2 + b(n)^2 = 13, with the companion sequence b(n)= A077236(n).
%D A054491 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7(1969), 
               pps. 181-193.
%D A054491 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, 
               pps. 122-125, 194-196.
%D A054491 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 
               7(1969), pps. 231-242.
%H A054491 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A054491 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A054491 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A054491 a(n)= (6*((2+sqrt(3))^n-(2-sqrt(3))^n)-((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)))/
               (2*sqrt(3)).
%F A054491 a(n)= 6*S(n-1, 4) - S(n-2, 4) = S(n, 4) + 2*S(n-1, 4), with S(n, x) := 
               U(n, x/2) Chebyshev's polynomials of 2nd kind, A049310. S(-1, x) 
               := 0, S(-2, x) := -1, S(n, 4)= A001353(n+1).
%F A054491 G.f.: (1+2*x)/(1-4*x+x^2).
%F A054491 Conjecture: a(n+1) = A001353(n+2) + 2*A001353(n+1) - Creighton Dement 
               (creighton.k.dement(AT)uni-oldenburg.de), Nov 28 2004. Comment from 
               Vim Wenders (vim(AT)gmx.li), Mar 26 2008: The conjecture is easily 
               verified using a(n) = (6*((2+sqrt(3))^n-(2-sqrt(3))^n)-((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)))/
               (2*sqrt(3)) and A001353(n)=[(2+sqrt(3))^n-(2-sqrt(3))^n]/(2*sqrt(3)).
%Y A054491 Cf. A001353, A001834.
%Y A054491 Sequence in context: A078798 A027043 A006815 this_sequence A013261 A013265 
               A038383
%Y A054491 Adjacent sequences: A054488 A054489 A054490 this_sequence A054492 A054493 
               A054494
%K A054491 easy,nonn
%O A054491 0,2
%A A054491 Barry E. Williams, May 04 2000
%E A054491 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 05 2000
%E A054491 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 08 2002

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