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A055243 First differences of A001628 (Fibonacci convolution). 6
1, 2, 6, 13, 29, 60, 122, 241, 468, 894, 1686, 3144, 5807, 10636, 19338, 34931, 62731, 112068, 199264, 352787, 622152, 1093260, 1914780, 3343440, 5821645, 10110278, 17515566, 30276073, 52221929, 89896332, 154461110, 264930661, 453654108, 775598634, 1324053522 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

2*a(n) = C_{n+3} of Turban reference eq.(2.17), C_{1}= 0 = C_{2}.

Number of binary sequences of length n+3 such that the sequence has exactly two pairs (which may overlap) of consecutive 1's. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 07 2004

REFERENCES

L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

FORMULA

G.f.: (1-x)/(1-x-x^2)^3. (from Turban reference eq.(2.15)).

a(n)= ((5*n^2+37*n+50)*F(n+1)+4*(n+1)*F(n))/50 with F(n)=A000045(n) (Fibonacci numbers) (from Turban reference eq. (2.17)).

Comments from Peter Bala, Oct 25 2007 (Start): Since F(-n) = (-1)^(n+1)*F(n), we can use the previous formula to extend the sequence to negative values of n; we find a(-n) = (-1)^n* A129707(n-3).

Recurrence relations: a(n+4)=2a(n+3)+a(n+2)-2a(n+1)-a(n)+F(n+3), a(0)=1, a(1)=2, a(2)=6, a(3)=13; a(n+2)=a(n+1)+a(n)+A010049(n+3), a(0)=1, a(1)=2.

a(n-3) = sum {k = 2..floor((n+1)/2)} C(k,2)C(n-k,k-1) = (1/2)*G''(n,1), where the polynomial G(n,x) := sum {k = 1..floor((n+1)/2)} C(n-k,k-1)* x^k = x^((n+1)/2) * F(n, 1/sqrt(x)) and where F(n,x) denotes the n-th Fibonacci polynomial. Since G(n,1) yields the Fibonacci numbers A000045 and G'(n,1) yields the second-order Fibonacci numbers A010049, a(n) may be considered as the sequence of third-order Fibonacci numbers.

For n >= 4, the polynomials sum {k = 0..n} C(n,k)* G''(n-k,1)*(-x)^k appear to satisfy a Riemann hypothesis; their zeros appear to lie on the vertical line Re x = 1/2 in the complex plane. Compare with the remarks in A094440 and A010049. (End)

MAPLE

a:= n -> (Matrix([[1, 0$4, -1]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -5, 0, 3, 1][i] else 0 fi)^(n))[1, 1]: seq(a(n), n=0..30); # Alois P. Heinz, Aug 05 2008

CROSSREFS

Cf. A001628, A000045.

Cf. A010049, A094440, A129707.

Sequence in context: A289525 A289764 A289887 * A173009 A212586 A276411

Adjacent sequences:  A055240 A055241 A055242 * A055244 A055245 A055246

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 10 2000

STATUS

approved

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Last modified October 17 01:24 EDT 2018. Contains 316275 sequences. (Running on oeis4.)