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A077409
Bisection (even part) of Chebyshev sequence with Diophantine property.
6
7, 59, 583, 5771, 57127, 565499, 5597863, 55413131, 548533447, 5429921339, 53750679943, 532076878091, 5267018100967, 52138104131579, 516114023214823, 5109002128016651, 50573907256951687, 500630070441500219, 4955726797158050503, 49056637901139004811
OFFSET
0,1
COMMENTS
a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).
The odd part is A077250(n) with Diophantine companion A077249(n).
FORMULA
a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.
a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).
a(n) = sqrt(24*A077251(n)^2 + 25).
G.f.: (7-11*x)/(1-10*x+x^2).
EXAMPLE
59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.
MATHEMATICA
CoefficientList[Series[(7 - 11 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {7, 59}, 30] (* G. C. Greubel, Jan 18 2018 *)
PROG
(PARI) a(n)=if(n<0, 0, subst(poltchebi(n+1)+2*poltchebi(n), x, 5))
(PARI) Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015
(PARI) a(n)=polchebyshev(n+1, , 5)+2*polchebyshev(n, , 5) \\ Charles R Greathouse IV, Jun 15 2015
(PARI) a(n)=([0, 1; -1, 10]^n*[7; 59])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015
(Magma) I:=[7, 59]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018
CROSSREFS
Sequence in context: A099659 A358599 A135150 * A192458 A203237 A099347
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 08, 2002
STATUS
approved