0,4

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 (deutsch(AT)duke.poly.edu), Dec 10 2003

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.

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 (benoit7848c(AT)orange.fr), 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 (pbala(AT)toucansurf.com), Oct 24 2007

a(n) = -A001629(n+2)+2*A001629(n+1)+A000045(n+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

Starting (1, 1, 3, 5, 10,...), = row sums of triangle A135830. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007

with(combinat): A010049 := proc(n) options remember; if n <= 1 then n else A010049(n-1)+A010049(n-2)+fibonacci(n-2); fi; end;

First differences of A001629.

Partial sums of A006367. A023610, A094440, A134410.

Cf. A135830.

Sequence in context: A009854 A018165 A054179 this_sequence A094986 A154949 A107232

Adjacent sequences: A010046 A010047 A010048 this_sequence A010050 A010051 A010052

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 03 2000

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003