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A077249
Bisection (odd part) of Chebyshev sequence with Diophantine property.
6
2, 21, 208, 2059, 20382, 201761, 1997228, 19770519, 195707962, 1937309101, 19177383048, 189836521379, 1879187830742, 18602041786041, 184141230029668, 1822810258510639, 18043961355076722, 178616803292256581, 1768124071567489088, 17502623912382634299
OFFSET
0,1
COMMENTS
-24*a(n)^2 + b(n)^2 = 25, with the companion sequence b(n) = A077250(n).
The even part is A077251(n) with Diophantine companion A077409(n).
FORMULA
a(n) = 10*a(n-1)- a(n-2), a(-1) := -1, a(0)=2.
a(n) = 2*S(n, 10)+S(n-1, 10), with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 10)= A004189(n+1).
G.f.: (2+x)/(1-10*x+x^2).
EXAMPLE
24*a(1)^2 + 25 = 24*21^2+25 = 10609 = 103^2 = A077250(1)^2.
MATHEMATICA
CoefficientList[Series[(z + 2)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {2, 21}, 40] (* Harvey P. Dale, Apr 08 2012 *)
PROG
(PARI) a(n)=if(n<0, 0, subst(-7*poltchebi(n)+11*poltchebi(n+1), x, 5)/24)
(PARI) a(n)=2*polchebyshev(n, 2, 5)+polchebyshev(n-1, 2, 5) \\ Charles R Greathouse IV, Jun 11 2011
(PARI) Vec((2+x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015
CROSSREFS
Sequence in context: A365061 A110253 A185634 * A068070 A085953 A225614
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 08 2002
STATUS
approved