|
%I #41 Mar 06 2024 16:39:52
%S 1,3,7,16,35,75,158,329,679,1392,2839,5767,11678,23589,47555,95720,
%T 192427,386451,775486,1555153,3117071,6245088,12507887,25044431,
%U 50135230,100345485,200812363,401821144,803960099,1608434427,3217700894,6436748057
%N Expansion of (1-x)(1+x)/((2x-1)(x^2+x-1)).
%C A floretion-generated sequence relating to Fibonacci numbers and powers of 2. The sequence results from a particular transform of the sequence A000079*(-1)^n (powers of 2).
%C Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[ ( 5'i + .5i' + .5'ii' + .5e)*( + .5j' + .5'kk' + .5'ki' + .5e ) ], 1vesforseq = A000079(n+1)*(-1)^(n+1), ForType: 1A. Identity used: jesfor = jesrightfor + jesleftfor
%H Vincenzo Librandi, <a href="/A104004/b104004.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2).
%F 4*a(n) = A008466(n+3) + A027973(n) (FAMP result).
%F Suggestions made by Superseeker: a(n+2) - a(n+1) - a(n) = A042950(n+1).
%F Coefficients of g.f.*(1-x)/(1+x) match A099036.
%F Coefficients of g.f./(1+x) match A027934.
%F Coefficients of g.f./(1-x^2) match A008466;
%F a(n) = A101220(3, 2, n+1) - A101220(3, 2, n). - _Ross La Haye_, Aug 05 2005
%F a(n) = 3*2^n - Fibonacci(n+3) = A221719(n) + 1. - _Ralf Stephan_, May 20 2007, _Hugo Pfoertner_, Mar 06 2024
%F a(n) = (3*2^n - (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5)))) / sqrt(5)). - _Colin Barker_, Aug 18 2017
%p with (combinat):a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=fibonacci(n-1)+2*a[n-1] od: seq(a[n], n=1..26); # _Zerinvary Lajos_, Mar 17 2008
%t LinearRecurrence[{3, -1, -2}, {1, 3, 7}, 80] (* _Vincenzo Librandi_, Aug 18 2017 *)
%o (Magma) [3*2^n-Fibonacci(n+3): n in [0..40]]; // _Vincenzo Librandi_, Aug 18 2017
%Y Cf. A000079, A008466, A027934, A042950, A099036, A221719.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Feb 24 2005
|
