A104004 Expansion of (1-x)(1+x)/((2x-1)(x^2+x-1)).

1, 3, 7, 16, 35, 75, 158, 329, 679, 1392, 2839, 5767, 11678, 23589, 47555, 95720, 192427, 386451, 775486, 1555153, 3117071, 6245088, 12507887, 25044431, 50135230, 100345485, 200812363, 401821144, 803960099, 1608434427, 3217700894, 6436748057

Offset

0,2

Comments

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).

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Formula

4*a(n) = A008466(n+3) + A027973(n) (FAMP result).

Suggestions made by Superseeker: a(n+2) - a(n+1) - a(n) = A042950(n+1).

Coefficients of g.f.*(1-x)/(1+x) match A099036.

Coefficients of g.f./(1+x) match A027934.

Coefficients of g.f./(1-x^2) match A008466;

a(n) = A101220(3, 2, n+1) - A101220(3, 2, n). - Ross La Haye, Aug 05 2005

a(n) = 3*2^n - Fibonacci(n+3) = A221719(n) + 1. - Ralf Stephan, May 20 2007, Hugo Pfoertner, Mar 06 2024

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

Maple

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

Mathematica

LinearRecurrence[{3, -1, -2}, {1, 3, 7}, 80] (* Vincenzo Librandi, Aug 18 2017 *)
CoefficientList[Series[(1-x)(1+x)/((2x-1)(x^2+x-1)), {x, 0, 40}], x] (* Harvey P. Dale, Oct 12 2024 *)

Prog

(Magma) [3*2^n-Fibonacci(n+3): n in [0..40]]; // Vincenzo Librandi, Aug 18 2017

Crossrefs

Cf. A000079, A008466, A027934, A042950, A099036, A221719.

Sequence in context: A268394 A238913 A133124 * A101509 A240741 A240742

Adjacent sequences: A104001 A104002 A104003 * A104005 A104006 A104007

Keyword

easy,nonn

Author

Creighton Dement, Feb 24 2005

Status

approved