# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a085449 Showing 1-1 of 1 %I A085449 #40 Jun 29 2023 18:21:55 %S A085449 0,1,2,8,24,80,256,832,2688,8704,28160,91136,294912,954368,3088384, %T A085449 9994240,32342016,104660992,338690048,1096024064,3546808320, %U A085449 11477712896,37142659072,120196169728,388962975744,1258710630400 %N A085449 Horadam sequence (0,1,4,2). %C A085449 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 A085449 Binomial transform is A001076. - _Paul Barry_, Aug 25 2003 %H A085449 Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000 %H A085449 F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014. %H A085449 Eric Weisstein, Horadam Sequence. %H A085449 Eric Weisstein, Fibonacci Number. %H A085449 Eric Weisstein, Pell Number. %H A085449 Eric Weisstein, Lucas Number. %H A085449 Eric Weisstein, Lucas Sequence. %H A085449 Index entries for linear recurrences with constant coefficients, signature (2, 4). %F A085449 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 A085449 From _Paul Barry_, Aug 25 2003: (Start) %F A085449 G.f.: x/(1-2*x-4*x^2). %F A085449 a(n) = sqrt(5)*((1+sqrt(5))^n - (1-sqrt(5))^n)/10. %F A085449 a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)5^k . (End) %F A085449 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 A085449 a(n) = 2^(n-1)*Fib(n). - _Paul Barry_, Mar 22 2004 %F A085449 Sum_{n>=1} 1/a(n) = A269991. - _Amiram Eldar_, Feb 01 2021 %F A085449 a(n) = -(-4)^n*a(-n) for all integer n. - _Michael Somos_, Mar 07 2021 %e A085449 a(4) = 24 because a(3) = 8, a(2) = 2, s = 2, r = 4 and (2 * 8) + (4 * 2) = 24. %e A085449 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 A085449 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 A085449 Table[2^(n-1)*Fibonacci[n], {n,0,50}] (* _G. C. Greubel_, Oct 08 2018 *) %o A085449 (PARI) vector(50, n, n--; 2^(n-1)*fibonacci(n)) \\ _G. C. Greubel_, Oct 08 2018 %o A085449 (Magma) [2^(n-1)*Fibonacci(n): n in [0..50]]; // _G. C. Greubel_, Oct 08 2018 %o A085449 (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 A085449 Cf. A024318, A000032, A000129, A001076, A085939, A269991. %Y A085449 Essentially the same as A063727. %K A085449 easy,nonn %O A085449 0,3 %A A085449 _Ross La Haye_, Aug 18 2003 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE