A005673 a(n) = F(n+2) - 2^[ (n+1)/2 ] - 2^[ n/2 ] + 1.

0, 0, 0, 0, 1, 2, 6, 11, 24, 42, 81, 138, 250, 419, 732, 1214, 2073, 3414, 5742, 9411, 15664, 25586, 42273, 68882, 113202, 184131, 301428, 489654, 799273, 1297118, 2112774, 3426275, 5571816, 9030858, 14668209, 23764602, 38563882, 62459555, 101285580, 164007278

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
• Index entries for linear recurrences with constant coefficients, signature (2,2,-5,0,2).

Crossrefs

Cf. A000045.

Programs

Maple

A005673:=-z**4/(z-1)/(z**2+z-1)/(-1+2*z**2); # Conjectured by Simon Plouffe in his 1992 dissertation.

Mathematica

LinearRecurrence[{2,2,-5,0,2},{0,0,0,0,1},40] (* Harvey P. Dale, Apr 22 2024 *)

Formula

G.f.: x^4/((1-x)*(1-x-x^2)*(1-2x^2)); a(n) = 2*a(n-1)+2*a(n-2)-5*a(n-3)+2*a(n-5); a(n+1) = Sum_{k=0..n} (2^floor(k/2)-1)*F(n-k). - Paul Barry, Jul 28 2004