A005674 - OEIS

%I M2837 #33 Apr 13 2022 13:25:17

%S 0,0,0,0,1,3,10,25,63,144,327,711,1534,3237,6787,14056,28971,59283,

%T 120894,245457,497167,1004256,2025199,4077007,8198334,16467597,

%U 33052491,66293208

%N a(n) = 2^(n-1) + 2^[ n/2 ] + 2^[ (n-1)/2 ] - F(n+3).

%C a(n) is the number of compositions of n where mixing of even and odd summands occurs. That is, at least one even summand is bracketed by two odd summands, or vice versa. - _Gregory L. Simay_, Jul 27 2016

%D R. K. Guy, personal communication.

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

%H R. K. Guy, <a href="/A005667/a005667.pdf">Letter to N. J. A. Sloane, 1987</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%F From _Gregory L. Simay_, Jul 27 2016: (Start)

%F If n=2k, then a(n) = 2^(n-1) - 2*A079289(n) + 2^(n/2 - 1) + F(n).

%F If n=2k-1, then a(n) = 2^(n-1) - 2*A079289(n) + F(n). (End)

%e 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).

%p A005674:=-z**4/(2*z-1)/(z**2+z-1)/(-1+2*z**2); # [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]

%Y Cf. A079289, A027558 divided by 2.

%K nonn,easy

%O 0,6

%A _N. J. A. Sloane_.