A007382 Number of strict (-1)st-order maximal independent sets in path graph.

0, 0, 3, 4, 11, 16, 32, 49, 87, 137, 231, 369, 608, 978, 1595, 2574, 4179, 6754, 10944, 17699, 28655, 46355, 75023, 121379, 196416, 317796, 514227, 832024, 1346267

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• R. Yanco and A. Bagchi, K-th order maximal independent sets in path and cycle graphs, J. Graph Theory, submitted, 1994.

Links

• R. Yanco, Letter and Email to N. J. A. Sloane, 1994

Crossrefs

Equals A054451(n+1) - 1.

Programs

Mathematica

Table[Sum[Binomial[n - i + 1, i], {i, Floor[(n - 1)/2]}], {n, 30}] (* or *)
Rest@ Abs@ CoefficientList[Series[x^3*(x^3 + 2 x^2 - x - 3)/((1 - x - x^2) (1 - x^2)^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 19 2017 *)

Formula

John W. Layman observes that if b(n) = 1+A007382(n) then b(n) = b(n-1) + 3b(n-2) - 2b(n-3) - 3b(n-4) + b(n-5) + b(n-6) for all 27 terms shown.

G.f.: x^3*(x^3+2x^2-x-3)/((1-x-x^2)*(1-x^2)^2).

a(n) = Sum_{i=1..floor((n-1)/2)} C(n-i+1, i). - Wesley Ivan Hurt, Sep 19 2017