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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
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).
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)
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
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
(PARI) a(n) = polchebyshev(n+1, 1, 2) + 2*polchebyshev(n, 1, 2); \\ Michel Marcus, Oct 13 2021
(Magma) R<x>:=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
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 March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)