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 A077236 a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11. 8
 4, 11, 40, 149, 556, 2075, 7744, 28901, 107860, 402539, 1502296, 5606645, 20924284, 78090491, 291437680, 1087660229, 4059203236, 15149152715, 56537407624, 211000477781, 787464503500, 2938857536219, 10967965641376 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n). Bisection (even part) of Chebyshev sequence with Diophantine property. The odd part is A077235(n) with Diophantine companion A077234(n). LINKS Luigi Cerlienco, Maurice Mignotte, and F. Piras, Suites récurrentes linéaires: Propriétés algébriques et arithmétiques, L'Enseignement Math., 33 (1987), 67-108. See Example 2, page 93. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (4,-1). FORMULA 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). G.f.: (4-5*x)/(1-4*x+x^2). a(n) = -(1/2)*sqrt(3)*(2-sqrt(3))^n + (1/2)*sqrt(3)*(2+sqrt(3))^n + 2*(2-sqrt(3))^n + 2*(2+sqrt(3))^n, with n >= 0. - Paolo P. Lava, Nov 20 2008 From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009: (Start) a(n) = ((4+sqrt(3))*(2+sqrt(3))^n + (4-sqrt(3))*(2-sqrt(3))^n)/2. Offset 0. a(n) = second binomial transform of 4,3,12,9,36. (End) EXAMPLE 11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11. MATHEMATICA 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 *) PROG (PARI) my(x='x+O('x^30)); Vec((4-5*x)/(1-4*x+x^2)) \\ G. C. Greubel, Apr 28 2019 (MAGMA) R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // G. C. Greubel, Apr 28 2019 (Sage) ((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019 (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 CROSSREFS Cf. A077238 (even and odd parts), A077235, A053120. Sequence in context: A149267 A149268 A214142 * A327025 A228190 A289283 Adjacent sequences:  A077233 A077234 A077235 * A077237 A077238 A077239 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 08 2002 EXTENSIONS Edited by N. J. A. Sloane, Sep 07 2018, replacing old definition with simple formula from Philippe Deléham, Nov 16 2008 STATUS approved

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Last modified October 23 03:21 EDT 2019. Contains 328335 sequences. (Running on oeis4.)