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A077236
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Bisection (even part) of Chebyshev sequence with Diophantine property.
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8
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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; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n).
The odd part is A077235(n) with Diophantine companion A077234(n).
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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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)=4*a(n-1)-a(n-2) with a(0)=4 and a(1)=11. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2008]
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 [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 20 2008]
a(n)=((4+sqrt3)(2+sqrt3)^n+(4-sqrt3)(2-sqrt3)^n)/2. Offset 0. a(n)=second binomial transform of 4,3,12,9,36. [From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009]
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EXAMPLE
| 11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.
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CROSSREFS
| Cf. A077238 (even and odd parts).
Sequence in context: A149266 A149267 A149268 * A152532 A121096 A047091
Adjacent sequences: A077233 A077234 A077235 * A077237 A077238 A077239
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08, 2002
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