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Search: id:A077235
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| A077235 |
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Bisection (odd part) of Chebyshev sequence with Diophantine property. |
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+0
5
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| 5, 16, 59, 220, 821, 3064, 11435, 42676, 159269, 594400, 2218331, 8278924, 30897365, 115310536, 430344779, 1606068580, 5993929541, 22369649584, 83484668795, 311569025596, 1162791433589, 4339596708760
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A077234(n).
The even part is A077236(n) with Diophantine companion A054491(n).
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LINKS
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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
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a(n)= 2*T(n+1, 2)+T(n, 2), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 2)= A001075(n).
G.f.: (5-4*x)/(1-4*x+x^2).
a(n)=4*a(n-1)-a(n-2) with a(0)=5 and a(1)=16. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2008]
a(n)=-sqrt(3)*[2-sqrt(3)]^n+sqrt(3)*[2+sqrt(3)]^n+(5/2)*[2-sqrt(3)]^n+(5/2)*[2+sqrt(3)]^n, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
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EXAMPLE
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16 = a(1) = sqrt(3*A077234(1)^2 + 13) = sqrt(3*9^2 + 13)= sqrt(256) = 16.
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CROSSREFS
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Cf. A077238 (even and odd parts).
Adjacent sequences: A077232 A077233 A077234 this_sequence A077236 A077237 A077238
Sequence in context: A006217 A116914 A047103 this_sequence A098347 A034532 A092497
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
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