OFFSET
0,3
COMMENTS
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.
Binomial transform is A001076. - Paul Barry, Aug 25 2003
LINKS
Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Eric Weisstein, Horadam Sequence.
Eric Weisstein, Fibonacci Number.
Eric Weisstein, Pell Number.
Eric Weisstein, Lucas Number.
Eric Weisstein, Lucas Sequence.
Index entries for linear recurrences with constant coefficients, signature (2, 4).
FORMULA
a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 2, r = 4.
From Paul Barry, Aug 25 2003: (Start)
G.f.: x/(1-2*x-4*x^2).
a(n) = sqrt(5)*((1+sqrt(5))^n - (1-sqrt(5))^n)/10.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)5^k . (End)
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
a(n) = 2^(n-1)*Fib(n). - Paul Barry, Mar 22 2004
Sum_{n>=1} 1/a(n) = A269991. - Amiram Eldar, Feb 01 2021
a(n) = -(-4)^n*a(-n) for all integer n. - Michael Somos, Mar 07 2021
EXAMPLE
a(4) = 24 because a(3) = 8, a(2) = 2, s = 2, r = 4 and (2 * 8) + (4 * 2) = 24.
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
MAPLE
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
MATHEMATICA
Table[2^(n-1)*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Oct 08 2018 *)
PROG
(PARI) vector(50, n, n--; 2^(n-1)*fibonacci(n)) \\ G. C. Greubel, Oct 08 2018
(Magma) [2^(n-1)*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Oct 08 2018
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ross La Haye, Aug 18 2003
STATUS
approved