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A077251 Bisection (even part) of Chebyshev sequence with Diophantine property. 6
1, 12, 119, 1178, 11661, 115432, 1142659, 11311158, 111968921, 1108378052, 10971811599, 108609737938, 1075125567781, 10642645939872, 105351333830939, 1042870692369518, 10323355589864241, 102190685206272892, 1011583496472864679, 10013644279522373898 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
b(n)^2 - 24*a(n)^2 = 25, with the companion sequence b(n) = A077409(n).
The odd part is A077249(n) with Diophantine companion A077250(n).
LINKS
Tanya Khovanova, Recursive Sequences
FORMULA
a(n) = 10*a(n-1)- a(n-2), a(-1)=-2, a(0)=1.
a(n) = S(n, 10)+2*S(n-1, 10), with S(n, x) = U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310. S(n, 10)= A004189(n+1).
a(n) = sqrt((A077409(n)^2 - 25)/24).
G.f.: (1+2*x)/(1-10*x+x^2).
EXAMPLE
24*a(1)^2 + 25 = 24*12^2 + 25 = 3481 = 59^2 = A077409(1)^2.
MATHEMATICA
CoefficientList[Series[(2 z + 1)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
PROG
(PARI) Vec((1+2*x)/(1-10*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 11 2011
(PARI) a(n)=([0, 1; -1, 10]^n*[1; 12])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015
CROSSREFS
Sequence in context: A025132 A001712 A285232 * A289542 A075622 A153054
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 08 2002
STATUS
approved

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Last modified June 14 21:31 EDT 2024. Contains 373401 sequences. (Running on oeis4.)