Search: id:A054488
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%I A054488
%S A054488 1,8,47,274,1597,9308,54251,316198,1842937,10741424,62605607,364892218,
%T A054488 2126747701,12395593988,72246816227,421085303374,2454265004017,
%U A054488 14304504720728,83372763320351,485932075201378,2832219687887917
%N A054488 A second order recursive sequence.
%C A054488 Bisection (even part) of Chebyshev sequence with Diophantine property.
%C A054488 b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).
%C A054488 The odd part is A077413(n) with Diophantine companion A077239(n).
%D A054488 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7(1969), 
               pps. 181-193.
%D A054488 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, 
               pps. 122-125, 194-196.
%D A054488 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 
               7(1969), pps. 231-242.
%H A054488 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A054488 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A054488 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A054488 a(n)=6*a(n-1)-a(n-2), a(0)=1, a(1)=8.
%F A054488 a(n)=((3 + 2*sqrt(2))^(n+1) - (3 - 2*sqrt(2))^(n+1) + 2*((3 + 2*sqrt(2))^n 
               - (3 - 2*sqrt(2))^n))/(4*sqrt(2)).
%F A054488 a(n)= S(n, 6)+2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the 
               second kind, A049310. S(n, 6)= A001109(n+1).
%F A054488 G.f.: (1+2*x)/(1-6*x+x^2).
%e A054488 8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) 
               = 8.
%p A054488 a[0]:=1: a[1]:=8: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], 
               n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 
               2006
%Y A054488 Cf. A002315 and A038761.
%Y A054488 A077241 (even and odd parts).
%Y A054488 Sequence in context: A016198 A051140 A014524 this_sequence A034349 A024108 
               A165037
%Y A054488 Adjacent sequences: A054485 A054486 A054487 this_sequence A054489 A054490 
               A054491
%K A054488 easy,nonn
%O A054488 0,2
%A A054488 Barry E. Williams, May 04 2000
%E A054488 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 05 2000
%E A054488 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 08 2002

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