A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.
2, 21, 439, 9198, 192719, 4037901, 84603202, 1772629341, 37140612959, 778180242798, 16304644485799, 341619353958981, 7157701788652802, 149970118207749861, 3142214780574094279, 65836540273848229998
Offset: 0
References
- O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
Links
Crossrefs
Programs
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Mathematica
a[0] = 2; a[1] = 21; a[n_] := 21a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *) -
Sage
[lucas_number2(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008
Formula
a(n) = S(n, 21) - S(n-2, 21) = 2*T(n, 21/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 21)=A092499(n+1). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: (2-21*x)/(1-21*x+x^2).
Extensions
Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004
Comments