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%I #32 Sep 08 2022 08:45:11
%S 0,1,4,22,112,580,2992,15448,79744,411664,2125120,10970464,56632576,
%T 292353088,1509207808,7790949760,40219045888,207621882112,
%U 1071801803776,5532938507776,28562564853760
%N Horadam sequence (0,1,6,4).
%C a(n) / a(n-1) converges to sqrt(10) + 2 as n approaches infinity; sqrt(10) + 2 can also be written as sqrt(2) * (sqrt(2) + sqrt(5)), 2 * sqrt(2) * Phi - sqrt(2) + 2 and lim_{n->infinity} sqrt(2) * (sqrt(2) + (L(n) / F(n))), where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number.
%H G. C. Greubel, <a href="/A085939/b085939.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/HoradamSequence.html">Horadam Sequence</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/PellNumber.html">Pell Number</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, 6).
%F a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 6.
%F a(n) = ((2+sqrt(10))^n - (2-sqrt(10))^n)/(2*sqrt(10)). - _Rolf Pleisch_, Jul 06 2009
%F G.f.: x/(1-4*x-6*x^2). - _Colin Barker_, Jan 10 2012
%e a(4) = 112 because a(3) = 22, a(2) = 4, s = 4, r = 6 and (4 * 22) + (6 * 4) = 112.
%t Join[{a=0,b=1},Table[c=4*b+6*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)
%t LinearRecurrence[{4,6},{0,1},30] (* _Harvey P. Dale_, Jul 20 2016 *)
%o (Sage) [lucas_number1(n,4,-6) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-4*x-6*x^2))) \\ _G. C. Greubel_, Jan 16 2018
%o (Magma) I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1) + 6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 16 2018
%Y Cf. A024318, A000032, A000129.
%K easy,nonn
%O 0,3
%A _Ross La Haye_, Aug 16 2003