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%I #40 Jun 29 2023 18:21:55
%S 0,1,2,8,24,80,256,832,2688,8704,28160,91136,294912,954368,3088384,
%T 9994240,32342016,104660992,338690048,1096024064,3546808320,
%U 11477712896,37142659072,120196169728,388962975744,1258710630400
%N Horadam sequence (0,1,4,2).
%C a(n) / a(n-1) converges to sqrt(5) + 1 as n approaches infinity. sqrt(5) + 1 can also be written as Phi^3 - 1, 2 * Phi, Phi^2 + Phi - 1 and (L(n) / F(n)) + 1, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity.
%C Binomial transform is A001076. - _Paul Barry_, Aug 25 2003
%H Karl V. Keller, Jr., <a href="/A085449/b085449.txt">Table of n, a(n) for n = 0..1000</a>
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, Preprint on ResearchGate, March 2014.
%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 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, 4).
%F a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 2, r = 4.
%F From _Paul Barry_, Aug 25 2003: (Start)
%F G.f.: x/(1-2*x-4*x^2).
%F a(n) = sqrt(5)*((1+sqrt(5))^n - (1-sqrt(5))^n)/10.
%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)5^k . (End)
%F The signed version 0, 1, -2, ... has a(n)=sqrt(5)((sqrt(5)-1)^n-(-sqrt(5)-1)^n)/10. It is the second inverse binomial transform of A085449. - _Paul Barry_, Aug 25 2003
%F a(n) = 2^(n-1)*Fib(n). - _Paul Barry_, Mar 22 2004
%F Sum_{n>=1} 1/a(n) = A269991. - _Amiram Eldar_, Feb 01 2021
%F a(n) = -(-4)^n*a(-n) for all integer n. - _Michael Somos_, Mar 07 2021
%e a(4) = 24 because a(3) = 8, a(2) = 2, s = 2, r = 4 and (2 * 8) + (4 * 2) = 24.
%e G.f. = x + 2*x^2 + 8*x^3 + 24*x^4 + 80*x^5 + 256*x^6 + 832*x^7 + ... - _Michael Somos_, Mar 07 2021
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*(a[n-1]+2*a[n-2]) od: seq(a[n], n=0..26); # _Zerinvary Lajos_, Mar 17 2008
%t Table[2^(n-1)*Fibonacci[n], {n,0,50}] (* _G. C. Greubel_, Oct 08 2018 *)
%o (PARI) vector(50, n, n--; 2^(n-1)*fibonacci(n)) \\ _G. C. Greubel_, Oct 08 2018
%o (Magma) [2^(n-1)*Fibonacci(n): n in [0..50]]; // _G. C. Greubel_, Oct 08 2018
%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]+4*a[n-2]; od; a; # _Muniru A Asiru_, Oct 09 2018
%Y Cf. A024318, A000032, A000129, A001076, A085939, A269991.
%Y Essentially the same as A063727.
%K easy,nonn
%O 0,3
%A _Ross La Haye_, Aug 18 2003
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