A005686 Number of Twopins positions.

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 12, 14, 18, 22, 27, 34, 41, 52, 63, 79, 97, 120, 149, 183, 228, 280, 348, 429, 531, 657, 811, 1005, 1240, 1536, 1897, 2347, 2902, 3587, 4438, 5484, 6785, 8386, 10372, 12824, 15856, 19609, 24242, 29981, 37066, 45837

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Natasha Blitvić, Vicente I. Fernandez, A Combinatorial Model for Heterogeneous Microbial Growth, arXiv:1901.04080 [math.CO], 2019.
• S. Falcon, Generalized (k,r)-Fibonacci Numbers, Gen. Math. Notes, 25(2), 2014, 148-158.
• 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
• I. Wloch, U. Bednarz, D. Bród, A Wloch and M. Wolowiec-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Math., Volume 161, Issues 16-17, November 2013, Pages 2695-2701.
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 1).

Crossrefs

Cf. A001687.

Programs

Magma

I:=[1,1,1,1,1]; [0] cat [n le 5 select I[n] else Self(n-2)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 19 2016

Maple

A005686 := -(z+1)*(z**3+z+1)/(-1+z**2+z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1's
a := proc(n): if n = 0 then 0 else add(binomial(floor((n+3*k-4)/5), k), k=0..floor((n-1)/2)) fi: end: seq(a(n), n=0..54); # Johannes W. Meijer, Aug 05 2013

Mathematica

nn=54; CoefficientList[Series[(x+x^2)/(1-x^2-x^5),{x,0,nn}],x]  (* Geoffrey Critzer, Apr 28 2013 *)
m = 5; For[n = 0, n < m, n++, a[n] = 1]; For[n = m, n < 51, n++, a[n] = a[n - m] + a[n - 2]]; Table[a[n], {n, 0, 50}] (*Sergio Falcon, Nov 12 2015 *)
Join[{0}, LinearRecurrence[{0, 1, 0, 0, 1}, {1, 1, 1, 1, 1}, 60]] (* Vincenzo Librandi, Jan 19 2016 *)

PARI

a(n)=if(n<0,polcoeff((x^3+x^4)/(1+x^3-x^5)+x^-n*O(x),-n),polcoeff((x+x^2)/(1-x^2-x^5)+x^n*O(x),n)) /* Michael Somos, Jul 15 2004 */

PARI

a(n)=sum(k=0,(n-1)\2,binomial((n+3*k-4)\5,k))

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(floor((n+3k-3)/5), k). - Paul Barry, Jul 10 2004

G.f.: (x+x^2)/(1-x^2-x^5). - Ralf Stephan, Apr 21 2004

a(n) = A001687(n)+A001687(n-1). - Ralf Stephan, Apr 21 2004

a(n) = a(n-2) + a(n-5). - Michael Somos, Jul 15 2004

a(n+1) = Sum_{k=0..floor(n/5)} A065941(n-4*k, n-5*k). - Johannes W. Meijer, Aug 05 2013

Extensions

More terms from Paul Barry, Jul 10 2004