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%I #41 Jan 01 2024 11:05:31
%S 4,11,40,149,556,2075,7744,28901,107860,402539,1502296,5606645,
%T 20924284,78090491,291437680,1087660229,4059203236,15149152715,
%U 56537407624,211000477781,787464503500,2938857536219,10967965641376
%N a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.
%C a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n).
%C Bisection (even part) of Chebyshev sequence with Diophantine property.
%C The odd part is A077235(n) with Diophantine companion A077234(n).
%H Luigi Cerlienco, Maurice Mignotte, and F. Piras, <a href="http://dx.doi.org/10.5169/seals-87887">Suites récurrentes linéaires: Propriétés algébriques et arithmétiques</a>, L'Enseignement Math., 33 (1987), 67-108. See Example 2, page 93.
%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 a(n) = T(n+1,2) + 2*T(n,2), with T(n,x) Chebyshev's polynomials of the first kind, A053120. T(n,2) = A001075(n).
%F G.f.: (4-5*x)/(1-4*x+x^2).
%F From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009: (Start)
%F a(n) = ((4+sqrt(3))*(2+sqrt(3))^n + (4-sqrt(3))*(2-sqrt(3))^n)/2. Offset 0.
%F a(n) = second binomial transform of 4,3,12,9,36. (End)
%F a(n) = (A054491(n+1) - A054491(n-1))/2 = sqrt(3*A054491(n-1)*A054491(n+1) + 52), n >= 1. - _Klaus Purath_, Oct 12 2021
%e 11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.
%t CoefficientList[Series[(4-5*x)/(1-4*x+x^2), {x,0,20}], x] (* or *) LinearRecurrence[{4,-1}, {4,11}, 30] (* _G. C. Greubel_, Apr 28 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((4-5*x)/(1-4*x+x^2)) \\ _G. C. Greubel_, Apr 28 2019
%o (PARI) a(n) = polchebyshev(n+1, 1, 2) + 2*polchebyshev(n, 1, 2); \\ _Michel Marcus_, Oct 13 2021
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // _G. C. Greubel_, Apr 28 2019
%o (Sage) ((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 28 2019
%o (GAP) a:=[4,11];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Apr 28 2019
%Y Cf. A077238 (even and odd parts), A077235, A053120.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 08 2002
%E Edited by _N. J. A. Sloane_, Sep 07 2018, replacing old definition with simple formula from _Philippe Deléham_, Nov 16 2008
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