A005683 Numbers of Twopins positions. (Formerly M0695)

1, 2, 3, 5, 8, 13, 22, 37, 63, 108, 186, 322, 559, 973, 1697, 2964, 5183, 9071, 15886, 27835, 48790, 85545, 150021, 263136, 461596, 809812, 1420813, 2492945, 4374273, 7675598, 13468787, 23634817, 41474548, 72780553, 127718046, 224125677, 393308019, 690200668

Offset

3,2

Comments

Appears to be a bisection of A068930. - Ralf Stephan, Apr 20 2004

The Ze3 and Ze4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence with a(1) = 1 and a(2) = 1; the recurrence relation below confirms these values and gives a(0) = 0. - Johannes W. Meijer, Jul 14 2011

The complete sequence by R. K. Guy in "Anyone for Twopins?" starts with a(0)=0, a(1)=1 and a(2)=1 and has g.f. x*(1-x-x^2)/(1-2*x+x^4+x^6). - Johannes W. Meijer, Aug 14 2011

a(n) is the number of equivalence classes of subsets of {1..n-2} without isolated elements up to reflection. The reflection of a subset is the set obtained by mapping each element i to n + 1 - i. For example, the a(6)=5 equivalence classes of subsets of {1..4} are {}, {1,2}/{3,4}, {2,3}, {1,2,3}/{2,3,4}, {1,2,3,4}. If reflections are not considered equivalent then A005251(n) gives the number of subsets of {1..n-2} without isolated elements. - Andrew Howroyd, Dec 24 2019

References

R. K. Guy, "Anyone for Twopins?", in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.

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=3..40.

R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]

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, 0, 0, -1, 0, -1).

Formula

G.f.: x^3*(1-x^2-x^3-x^4-x^5)/(1-2*x+x^4+x^6). - Ralf Stephan, Apr 20 2004

a(3)=1, a(4)=2, a(5)=3, a(6)=5, a(7)=8, a(8)=13, a(n)=2*a(n-1)- a(n-4)- a(n-6). - Harvey P. Dale, Jun 20 2011

a(n) = (A005251(n) + A000931(n+4))/2. - Andrew Howroyd, Dec 24 2019

Maple

A005683:=-(-1+z**2+z**3+z**4+z**5)/(z**3-z**2+2*z-1)/(z**3+z**2-1); [Conjectured by Simon Plouffe in his 1992 dissertation.]

Mathematica

CoefficientList[Series[(1-x^2-x^3-x^4-x^5)/(1-2x+x^4+x^6), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 0, 0, -1, 0, -1}, {1, 2, 3, 5, 8, 13}, 40] (* Harvey P. Dale, Jun 20 2011 *)

Crossrefs

Cf. A000931, A005251.

Sequence in context: A306215 A018152 A293078 * A329698 A173404 A325473

Adjacent sequences: A005680 A005681 A005682 * A005684 A005685 A005686

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Jun 20 2011

Status

approved