1,3

For n>2 note that (n+1)|a(n) unless n is prime, in which case (n+1)|2*a(n). This sequence is not the better-known generalized Lucas numbers V(n,a,b) defined for fixed integers a and b such that D = a^2 + 4*b is nonnegative, V(0) = 2, V(1) = a and for n>1 the recurrence V(n) = V(n-1) + V(n-2). The a = b = 1 case gives the Lucas Numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 16 2005

Number of circular binary words of length n+1 having exactly two occurrences of 00. Example: a(4)=5 because we have 00011, 10001, 11000, 00110 and 01100. Column 2 of A119458. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 20 2006

L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fib. Quart., 15 (1977), 246-254.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

G.f.: x(1-x)(1-2x+2x^2)/(1-x-x^2)^3. - Ralf Stephan, Apr 23 2004, corrected Feb 08 2006

a(n)=a(n-1)+a(n-2)+n*Fibonacci(n-2)-(n-1)*Fibonacci(n-3) for n>=3; a(1)=1, a(2)=0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 20 2006

G:=x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3: Gser:=series(G, x=0, 45): seq(coeff(Gser, x^n), n=1..40); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 07 2006

with(combinat): a[1]:=1: a[2]:=0: for n from 3 to 40 do a[n]:=a[n-1]+a[n-2]+n*fibonacci(n-2)-(n-1)*fibonacci(n-3) od: seq(a[n], n=1..40); - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 20 2006

A006491:=(z-1)*(1-2*z+2*z**2)/(z**2+z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A006490.

Cf. A119458.

Sequence in context: A084179 A026634 A026656 this_sequence A051721 A050226 A119562

Adjacent sequences: A006488 A006489 A006490 this_sequence A006492 A006493 A006494

nonn,easy

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

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 07 2006