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%I #48 Sep 08 2022 08:45:01
%S 1,6,23,86,321,1198,4471,16686,62273,232406,867351,3236998,12080641,
%T 45085566,168261623,627960926,2343582081,8746367398,32641887511,
%U 121821182646,454642843073,1696750189646,6332357915511,23632681472398
%N a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(1)=6.
%C Bisection (even part) of Chebyshev sequence with Diophantine property.
%C The odd part is A077234 with Diophantine companion A077235.
%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="/A054491/b054491.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 (4,-1).
%F -3*a(n)^2 + A077236(n)^2 = 13.
%F 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 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 G.f.: (1+2*x)/(1-4*x+x^2).
%F a(n+1) = A001353(n+2) + 2*A001353(n+1) - _Creighton Dement_, Nov 28 2004. Comment from _Vim Wenders_, Mar 26 2008: This 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)).
%F a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-7)^k. - _Philippe Deléham_, Mar 05 2014
%F E.g.f.: (1/3)*exp(2*x)*(3*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x)). - _Stefano Spezia_, Jan 27 2020
%p seq( simplify(ChebyshevU(n,2) +2*ChebyshevU(n-1,2)), n=0..30); # _G. C. Greubel_, Jan 15 2020
%t Table[ChebyshevU[n, 2] +2*ChebyshevU[n-1, 2], {n,0,30}] (* _G. C. Greubel_, Jan 15 2020 *)
%t LinearRecurrence[{4,-1},{1,6},30] (* _Harvey P. Dale_, Sep 04 2021 *)
%o (PARI) a(n) = if (n==0, 1, if (n==1, 6, 4*a(n-1)-a(n-2))) \\ _Michel Marcus_, Jun 23 2013
%o (PARI) a(n) = polchebyshev(n, 2, 2) + 2*polchebyshev(n-1, 2, 2); \\ _Michel Marcus_, Oct 13 2021
%o (Magma) I:=[1,6]; [n le 2 select I[n] else 4*Self(n-1) -Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 15 2020
%o (Sage) [chebyshev_U(n,2) +2*chebyshev_U(n-1,2) for n in (0..30)]; # _G. C. Greubel_, Jan 15 2020
%o (GAP) a:=[1,6];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 15 2020
%Y Cf. A001353, A001834, A077234, A077235.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 04 2000
%E Chebyshev comments from _Wolfdieter Lang_, Nov 08 2002
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