A005674 a(n) = 2^(n-1) + 2^[ n/2 ] + 2^[ (n-1)/2 ] - F(n+3).
0, 0, 0, 0, 1, 3, 10, 25, 63, 144, 327, 711, 1534, 3237, 6787, 14056, 28971, 59283, 120894, 245457, 497167, 1004256, 2025199, 4077007, 8198334, 16467597, 33052491, 66293208
Offset: 0
Examples
a(6) = a(2*3) = 2^5 - f(9) + 3*2^2 = 32 - 34 + 12 = 10. The 10 compositions are (1,4,1), (3,2,1), (1,2,3), (2,1,2,1), (1,2,1,2), (2,1,1,2), (1,2,2,1), (1,2,1,1,1), (1,1,2,1,1), (1,1,1,2,1).
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
Programs
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Maple
A005674:=-z**4/(2*z-1)/(z**2+z-1)/(-1+2*z**2); # [Conjectured by Simon Plouffe in his 1992 dissertation.]
Formula
From Gregory L. Simay, Jul 27 2016: (Start)
If n=2k, then a(n) = 2^(n-1) - 2*A079289(n) + 2^(n/2 - 1) + F(n).
If n=2k-1, then a(n) = 2^(n-1) - 2*A079289(n) + F(n). (End)
Comments