A005674 a(n) = 2^(n-1) + 2^[ n/2 ] + 2^[ (n-1)/2 ] - F(n+3). (Formerly M2837)

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

0,6

Comments

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

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

Table of n, a(n) for n=0..27.

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

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)

Example

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).

Maple

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

Crossrefs

Cf. A079289, A027558 divided by 2.

Sequence in context: A067988 A297186 A262380 * A089100 A089117 A176610

Adjacent sequences: A005671 A005672 A005673 * A005675 A005676 A005677

Keyword

nonn,easy

Author

N. J. A. Sloane.

Status

approved