A085449 Horadam sequence (0,1,4,2).

0, 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136, 294912, 954368, 3088384, 9994240, 32342016, 104660992, 338690048, 1096024064, 3546808320, 11477712896, 37142659072, 120196169728, 388962975744, 1258710630400

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

Cf. A024318, A000032, A000129, A001076, A085939, A269991.

Essentially the same as A063727.

Sequence in context: A327550 A034741 A063727 * A127362 A133443 A094038

Adjacent sequences: A085446 A085447 A085448 * A085450 A085451 A085452

Keyword

easy,nonn

Author

Ross La Haye, Aug 18 2003

Status

approved