A041126 Numerators of continued fraction convergents to sqrt(72).
8, 17, 280, 577, 9512, 19601, 323128, 665857, 10976840, 22619537, 372889432, 768398401, 12667263848, 26102926097, 430314081400, 886731088897, 14618011503752, 30122754096401, 496582077046168, 1023286908188737, 16869172608065960, 34761632124320657
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
- Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).
Programs
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Mathematica
Numerator[Convergents[Sqrt[72], 30]] (* Vincenzo Librandi, Oct 29 2013 *) a0[n_] := (-4+3*Sqrt[2])*(17+12*Sqrt[2])^n-((4+3*Sqrt[2])/(17+12*Sqrt[2])^n) // Simplify a1[n_] := (1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/2 // FullSimplify Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
Formula
G.f.: -(x+1)*(x^2-9*x-8) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 05 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a1(n),a0(n)] for n>0:
a0(n) = (-4+3*sqrt(2))*(17+12*sqrt(2))^n-((4+3*sqrt(2))/(17+12*sqrt(2))^n).
a1(n) = (1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/2. (End)
Extensions
More terms from Colin Barker, Nov 05 2013