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
Examples
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
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).
Crossrefs
Programs
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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
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Magma
[2^(n-1)*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Oct 08 2018
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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
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Mathematica
Table[2^(n-1)*Fibonacci[n], {n,0,50}] (* G. C. Greubel, Oct 08 2018 *) -
PARI
vector(50, n, n--; 2^(n-1)*fibonacci(n)) \\ G. C. Greubel, Oct 08 2018
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
Comments