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A129707 Number of inversions in all Fibonacci binary words of length n. 7
0, 0, 1, 4, 12, 31, 73, 162, 344, 707, 1416, 2778, 5358, 10188, 19139, 35582, 65556, 119825, 217487, 392286, 703618, 1255669, 2230608, 3946020, 6954060, 12212280, 21377365, 37309288, 64935132, 112726771, 195224773, 337343034, 581700476 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A Fibonacci binary word is a binary word having no 00 subword.

LINKS

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

FORMULA

a(n) = Sum_{k>=0} k*A129706(n,k).

G.f.: z^2*(1+z)/(1-z-z^2)^3.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) + F(n), a(0)=a(1)=0, a(2)=1, a(3)=4.

a(n-3) = ((5*n^2 - 37*n + 50)*F(n-1) + 4*(n-1)*F(n))/50 = (-1)^n*A055243(-n). - Peter Bala, Oct 25 2007

a(n) = A001628(n-3) + A001628(n-2). - R. J. Mathar, Dec 07 2011

a(n+1) = A123585(n+2,n). - Philippe Deléham, Dec 18 2011

a(n) = Sum_{k=floor((n-1)/2)..n-1}  k*(k+1)/2*C(k,n-k-1). - Vladimir Kruchinin, Sep 17 2020

EXAMPLE

a(3)=4 because the Fibonacci words 110,111,101,010,011 have a total of 2 + 0 + 1 + 1 + 0 = 4 inversions.

MAPLE

with(combinat): a[0]:=0: a[1]:=0: a[2]:=1: a[3]:=4: for n from 4 to 40 do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]-a[n-4]+fibonacci(n) od: seq(a[n], n=0..40);

MATHEMATICA

CoefficientList[Series[x^2*(1 + x)/(1 - x - x^2)^3, {x, 0, 50}], x] (* G. C. Greubel, Mar 04 2017 *)

PROG

(PARI) x='x+O('x^50); concat([0, 0], Vec(x^2*(1 + x)/(1 - x - x^2)^3)) \\ G. C. Greubel, Mar 04 2017

(Maxima)

a(n) = sum(k*(k+1)*binomial(k, n-k-1), k, floor((n-1)/2), n-1)/2; /* Vladimir Kruchinin, Sep 17 2020 */

CROSSREFS

Cf. A129706.

Cf. A055243.

Sequence in context: A037255 A027658 A001982 * A320545 A232580 A133546

Adjacent sequences:  A129704 A129705 A129706 * A129708 A129709 A129710

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 12 2007

STATUS

approved

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Last modified July 28 17:47 EDT 2021. Contains 346335 sequences. (Running on oeis4.)