|
%I #27 Dec 07 2019 12:18:24
%S 2,23,527,12098,277727,6375623,146361602,3359941223,77132286527,
%T 1770682648898,40648568638127,933146396028023,21421718540006402,
%U 491766380024119223,11289205022014735727,259159949126314802498
%N a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.
%C A Chebyshev T-sequence with Diophantine property.
%C a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 21*(5*b)^2 =+4 with companion sequence b(n)=A097778(n-1), n>=1; b(0):=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="/A090731/b090731.txt">Table of n, a(n) for n = 0..733</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 (23,-1).
%F a(n) = S(n, 23) - S(n-2, 23) = 2*T(n, 23/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 23)=A097778(n). 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 := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.
%F G.f.: (2-23*x)/(1-23*x+x^2).
%e (x;y) = (0;2), (23;1), (527;23), (12098;528), ... give the
%e nonnegative integer solutions to x^2 - 21*(5*y)^2 = 4.
%t a[0] = 2; a[1] = 23; a[n_] := 23a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t LinearRecurrence[{23,-1},{2,23},30] (* _Harvey P. Dale_, Feb 20 2012 *)
%o (Sage) [lucas_number2(n,23,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 26 2008
%Y Cf. A037088, A051502.
%Y a(n)=sqrt(4 + 21*(5*A097778(n-1))^2), n>=1.
%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
|
