A090729 - OEIS

%I #23 Dec 07 2019 12:18:24

%S 2,21,439,9198,192719,4037901,84603202,1772629341,37140612959,

%T 778180242798,16304644485799,341619353958981,7157701788652802,

%U 149970118207749861,3142214780574094279,65836540273848229998

%N a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

%C A Chebyshev T-sequence with Diophantine property.

%C a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 437*b^2 =+4 with companion sequence b(n)=A092499(n), n>=0.

%D O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

%H Indranil Ghosh, <a href="/A090729/b090729.txt">Table of n, a(n) for n = 0..755</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (21,-1).

%F 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.

%F a(n) = ap^n + am^n, with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.

%F G.f.: (2-21*x)/(1-21*x+x^2).

%t 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 *)

%o (Sage) [lucas_number2(n,21,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 27 2008

%Y Cf. A085985.

%Y a(n)=sqrt(4 + 437*A092499(n)^2), n>=1, (Pell equation d=437, +4).

%Y Cf. A077428, A078355 (Pell +4 equations).

%K easy,nonn

%O 0,1

%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

%E Chebyshev and Pell comments from _Wolfdieter Lang_, Sep 10 2004