A005672 a(n) = Fibonacci(n+1) - 2^floor(n/2).

0, 0, 0, 1, 1, 4, 5, 13, 18, 39, 57, 112, 169, 313, 482, 859, 1341, 2328, 3669, 6253, 9922, 16687, 26609, 44320, 70929, 117297, 188226, 309619, 497845, 815656, 1313501, 2145541

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• R. K. Guy, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• R. K. Guy, Letter to N. J. A. Sloane, 1987
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Crossrefs

Gives diagonal sums of triangle A054143.

Programs

Maple

A005672:=z**3/(z**2+z-1)/(-1+2*z**2); # conjectured by Simon Plouffe in his 1992 dissertation
with (combinat):a[ -1]:=0:a[1]:=0:a[2]:=1:for n from 2 to 50 do a[n]:=fibonacci(n-1)+2*a[n-2] od: seq(a[n-1], n=0..31); # Zerinvary Lajos, Mar 17 2008

Formula

a(n) = Fibonacci(n-1) + 2*a(n-2), a(-1)=0, a(1)=0, a(2)=1. - Zerinvary Lajos, Mar 17 2008