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A010049
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Second-order Fibonacci numbers.
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16
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0, 1, 1, 3, 5, 10, 18, 33, 59, 105, 185, 324, 564, 977, 1685, 2895, 4957, 8462, 14406, 24465, 41455, 70101, 118321, 199368, 335400, 563425, 945193, 1583643, 2650229, 4430290, 7398330, 12342849, 20573219, 34262337, 57013865, 94800780, 157517532, 261545777
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OFFSET
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0,4
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COMMENTS
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Number of parts in all compositions of n+1 with no 1's. E.g. a(5)=10 because in the compositions of 6 with no part equal to 1, namely 6,4+2,3+3,2+4,2+2+2, the total number of parts is 10. - Emeric Deutsch, Dec 10 2003
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REFERENCES
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D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 83.
L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.
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LINKS
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Table of n, a(n) for n=0..37.
Index to sequences with linear recurrences with constant coefficients, signature (2,1,-2,-1)
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FORMULA
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a(n)=((2*n+3)*F(n)-n*F(n-1))/5, F(n)=A000045(n) (Fibonacci numbers) (Turban reference eq.(2.12)).
G.f.: x*(1-x)/(1-x-x^2)^2 (Turban reference eq.(2.10)).
Recurrence : a(0)=0 a(1)=1 a(2)=1 a(n+2)=a(n+1)+a(n)+F(n) - Benoit Cloitre, Sep 02 2002
Set A(n) = a(n+1) + a(n-1), B(n) = a(n+1) - a(n-1). Then A(n+2) = A(n+1) + A(n) + Lucas(n) and B(n+2) = B(n+1) + B(n) + Fibonacci(n). The polynomials F_2(n,-x) = sum {k = 0..n} C(n,k)*a(n-k)*(-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 polynomials F(n,-x) defined in A094440. For a similar conjecture for polynomials involving the second-order Lucas numbers see A134410. - Peter Bala, Oct 24 2007
a(n) = -A001629(n+2)+2*A001629(n+1)+A000045(n+1). - R. J. Mathar, Nov 16 2007
Starting (1, 1, 3, 5, 10,...), = row sums of triangle A135830. - Gary W. Adamson, Nov 30 2007
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MAPLE
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with(combinat): A010049 := proc(n) options remember; if n <= 1 then n else A010049(n-1)+A010049(n-2)+fibonacci(n-2); fi; end;
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MATHEMATICA
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CoefficientList[Series[(z - z^2)/(z^2 + z - 1)^2, {z, 0, 100}], z] (* From Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
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CROSSREFS
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First differences of A001629.
Partial sums of A006367. A023610, A094440, A134410.
Cf. A135830.
Sequence in context: A009854 A018165 A054179 * A094986 A154949 A107232
Adjacent sequences: A010046 A010047 A010048 * A010050 A010051 A010052
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Additional comments from Wolfdieter Lang, May 03 2000
More terms from Emeric Deutsch, Dec 10 2003
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STATUS
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approved
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