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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A035615 Number of winning length n strings with a 2 symbol alphabet in "same game". 12
1, 0, 2, 2, 6, 12, 26, 58, 126, 278, 602, 1300, 2774, 5878, 12350, 25778, 53470, 110332, 226610, 463602, 945214, 1921550, 3896642, 7885092, 15927086, 32121582, 64697726 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Strings that can be reduced to null string by repeatedly removing an entire run of two or more consecutive symbols.

LINKS

Table of n, a(n) for n=0..26.

C. Burns and B. Purcell, A note on Stephan's conjecture 77, preprint, 2005.

Sascha Kurz, Polynomials for same game, pdf

Index entries for linear recurrences with constant coefficients, signature (4, -2, -8, 6, 6, -3, -2).

FORMULA

G.f.: x(2x^6-6x^5+8x^4+2x^3-6x^2+2x)/[(1-x^2)(1-2x)(1-x-x^2)^2] (conjectured). - Ralf Stephan, May 11 2004. Established by Burns and Purcell - see link.

a(0)=1, a(1)=0, a(2)=2, a(3)=2, a(4)=6, a(5)=12, a(6)=26, a(7)=58, a(n)=4*a(n-1)-2*a(n-2)-8*a(n-3)+6*a(n-4)+6*a(n-5)-3*a(n-6)-2*a(n-7). - Harvey P. Dale, Sep 26 2012

EXAMPLE

11011001 is a winning string since 110{11}001->11{000}1->{111}->null

MATHEMATICA

Join[{1}, Rest[CoefficientList[Series[x (2x^6-6x^5+8x^4+2x^3-6x^2+2x)/ ((1-x^2)(1-2x)(1-x-x^2)^2), {x, 0, 40}], x]]] (* or *) Join[{1}, LinearRecurrence[ {4, -2, -8, 6, 6, -3, -2}, {0, 2, 2, 6, 12, 26, 58}, 40]] (* Harvey P. Dale, Sep 26 2012 *)

PROG

(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1; -2, -3, 6, 6, -8, -2, 4]^(n-1)*[0; 2; 2; 6; 12; 26; 58])[1, 1], 1) \\ Charles R Greathouse IV, Jun 15 2015

CROSSREFS

Cf. A035617.

Sequence in context: A275439 A173392 A217211 * A115962 A019311 A216215

Adjacent sequences:  A035612 A035613 A035614 * A035616 A035617 A035618

KEYWORD

nonn,nice,easy

AUTHOR

Erich Friedman

EXTENSIONS

More terms from Naohiro Nomoto, Jul 09 2001

Further terms from Sascha Kurz, Oct 19 2001

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 18 23:26 EST 2018. Contains 299330 sequences. (Running on oeis4.)