login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005683 Numbers of Twopins positions.
(Formerly M0695)
3
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 (list; graph; refs; listen; history; text; internal format)
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

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (2, 0, 0, -1, 0, -1).

FORMULA

G.f.: (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

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

Sequence in context: A124429 A018152 A293078 * A173404 A213710 A288382

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 21 12:51 EST 2018. Contains 299411 sequences. (Running on oeis4.)