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A085449 Horadam sequence (0,1,4,2). 10
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 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, March 2014; Preprint on ResearchGate.

LINKS

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Eric Weisstein, Horadam Sequence

Eric Weisstein, Fibonacci Number

Eric Weisstein, Pell Number

Eric Weisstein, Lucas Number

Eric Weisstein, Lucas Sequence

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

G.f.: x/(1-2x-4x^2); a(n)=sqrt(5)((1+sqrt(5))^n-(1-sqrt(5))^n)/10; a(n)=sum{k=0..floor(n/2), C(n, 2k+1)5^k }. - Paul Barry, Aug 25 2003

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

EXAMPLE

a(4) = 24 because a(3) = 8, a(2) = 2, s = 2, r = 4 and (2 * 8) + (4 * 2) = 24.

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

Join[{a=0, b=1}, Table[c=2*b+4*a; a=b; b=c, {n, 50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 26 2011*)

CROSSREFS

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

Essentially the same as A063727.

Sequence in context: A006952 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

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Last modified August 19 01:22 EDT 2017. Contains 290786 sequences.