0,3

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023

G.f.: 1/(-1+x+x^2)/(-1+x^2)

Recurrence: a(0)=1, a(1)=1, a(2)=3, a(n) = 2*a(n-2) + a(n-3) + 1.

Sum(1/5*(3+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+_Z^2))+Sum(-1/2*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))

a(2*k)=sum(F(2*j+1), j=0..k)= F(2*(k+1)), k >= 0; a(2*k-1)=sum(F(2*j), j=0..k)= F(2*k+1)-1, k >= 1; F(n)=A000045(n) (Fibonacci numbers).

a(n) = a(n-1) + a(n-2) + (1+(-1)^n)/2. a(2n)=F(2n+2), a(2n+1)=F(2n+3)-1.

a(n)=sum{k=0..floor(n/2), binomial(n-k+1, k)} - Paul Barry (pbarry(AT)wit.ie), Oct 23 2004

a(n) = floor(phi^(n+2) / sqrt(5)), where phi is the golden ratio: phi = (1+sqrt(5))/2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 19 2005

a(n) =fibonacci(n-1)+a(n-2), a(2)=1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 17 2008

Another recurrence (with a different offset) is given in the Maple code.

spec := [S, {S=Prod(Sequence(Union(Prod(Z, Z), Z)), Sequence(Prod(Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

with (combinat):a[2]:=1:for n from 2 to 50 do a[n]:=fibonacci(n-1)+a[n-2] od: seq(a[n-1], n=3..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 17 2008

f:=proc(n) option remember; local t1; if n <= 2 then RETURN(1); fi: if n mod 2 = 1 then t1:=1 else t1:=0; fi: f(n-1)+f(n-2)+t1; end; [seq(f(n), n=1..100)]; - njas, May 25 2008

(PARI) a(n)=if(n<0, 0, fibonacci(n+2)-n%2)

a(n)= A054450(n+1, 1) (second column of triangle). Cf. A062114.

Cf. A074331.

Essentially the same as A074331.

Sequence in context: A064188 A088953 A025034 this_sequence A074331 A033955 A049720

Adjacent sequences: A052949 A052950 A052951 this_sequence A052953 A052954 A052955

easy,nonn

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Additional formulae and more terms from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 02 2000. Better description from Olivier Gerard (ogerard(AT)ext.jussieu.fr), Jun 05 2001.

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000