A133586 Expansion of x*(1+2*x)/( (x^2-x-1)*(x^2+x-1) ).
1, 2, 3, 6, 8, 16, 21, 42, 55, 110, 144, 288, 377, 754, 987, 1974, 2584, 5168, 6765, 13530, 17711, 35422, 46368, 92736, 121393, 242786, 317811, 635622, 832040, 1664080, 2178309, 4356618, 5702887, 11405774, 14930352, 29860704, 39088169, 78176338, 102334155
Offset: 1
Examples
a(5) = F(6) = 8. a(6) = 2*a(5) = 2*8 = 16.
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).
Programs
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Maple
A133586aux := proc(n,k) add(A133080(n,j)*A133566(j,k),j=k..n) ; end proc: A000045 := proc(n) combinat[fibonacci](n) ; end proc: A133586 := proc(n) add(A133586aux(n,j)*A000045(j),j=0..n) ; end proc: # R. J. Mathar, Jun 20 2015
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Mathematica
CoefficientList[Series[(1 + 2 x)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 21 2015 *) LinearRecurrence[{0,3,0,-1},{1,2,3,6},40] (* Harvey P. Dale, Dec 10 2017 *) -
PARI
{a(n) = if( n%2, fibonacci(n+1), 2*fibonacci(n))}; /* Michael Somos, Jun 20 2015 */ -
PARI
Vec(x*(1+2*x)/((x^2-x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Mar 28 2016
Formula
Equals A133080 * A133566 * A000045, where A133080 and A133566 are infinite lower triangular matrices and the Fibonacci sequence as a vector (previous definition).
For odd-indexed terms, a(n) = F(n+1). For even-indexed terms, a(n) = 2*a(n-1).
a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-5+sqrt(5)) - (-1-sqrt(5))^n*(-3+sqrt(5)) - (-1+sqrt(5))^n*(3+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5))))/sqrt(5). - Colin Barker, Mar 28 2016
Extensions
New definition and A-number in previous definition corrected by R. J. Mathar, Jun 20 2015
Comments