|
%I #44 Sep 08 2022 08:45:01
%S 1,8,47,274,1597,9308,54251,316198,1842937,10741424,62605607,
%T 364892218,2126747701,12395593988,72246816227,421085303374,
%U 2454265004017,14304504720728,83372763320351,485932075201378,2832219687887917
%N Expansion of (1+2*x)/(1-6*x+x^2).
%C Bisection (even part) of Chebyshev sequence with Diophantine property.
%C b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).
%C The odd part is A077413(n) with Diophantine companion A077239(n).
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
%H G. C. Greubel, <a href="/A054488/b054488.txt">Table of n, a(n) for n = 0..1000</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=8.
%F 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 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 a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-9)^k. - _Philippe Deléham_, Mar 05 2014
%F a(n) = (Pell(2*n+2) + 2*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + Pell(2*n))/2. - _G. C. Greubel_, Jan 19 2020
%F E.g.f.: (1/4)*exp(3*x)*(4*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x)). - _Stefano Spezia_, Jan 27 2020
%e 8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.
%p 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_, Jul 26 2006
%t LinearRecurrence[{6,-1},{1,8},30] (* _Harvey P. Dale_, Oct 09 2017 *)
%t Table[(LucasL[2*n+1, 2] + Fibonacci[2*n, 2])/2, {n,0,30}] (* _G. C. Greubel_, Jan 19 2020 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+2*x)/(1-6*x+x^2)) \\ _G. C. Greubel_, Jan 19 2020
%o (PARI) apply( {A054488(n)=[1,8]*([0,-1;1,6]^n)[,1]}, [0..30]) \\ _M. F. Hasler_, Feb 27 2020
%o (Magma) I:=[1,8]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 19 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 21); Coefficients(R!( (1+2*x)/(1-6*x+x^2))); // _Marius A. Burtea_, Jan 20 2020
%o (Sage) [(lucas_number2(2*n+1,2,-1) + lucas_number1(2*n,2,-1))/2 for n in (0..30)] # _G. C. Greubel_, Jan 19 2020
%o (GAP) a:=[1,8];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 19 2020
%Y Cf. A000129, A002203, A002315, A038761, A077239, A077240, A077413.
%Y Cf. A077241 (even and odd parts).
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 04 2000
%E More terms from _James A. Sellers_, May 05 2000
%E Chebyshev comments from _Wolfdieter Lang_, Nov 08 2002
| 