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A030436 Expansion of (1+x-2*x^2-x^3)/(1-4*x^2+2*x^4). 13
1, 1, 2, 3, 6, 10, 20, 34, 68, 116, 232, 396, 792, 1352, 2704, 4616, 9232, 15760, 31520, 53808, 107616, 183712, 367424, 627232, 1254464, 2141504, 4283008, 7311552, 14623104, 24963200, 49926400, 85229696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also (starting 3, 6, ...) the number of zig-zag paths from top to bottom of a rectangle of width 7 whose color is not that of the top right corner.

From Johannes W. Meijer, May 29 2010: (Start)

The a(n) represent the number of possible chess games, ignoring the fifty-move and the triple repetition rules, after n moves by White in the following position: White Ka1, Nh1, pawns a2, b6, c2, d6, f2, g3 and h2; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3, g4 and h3.

Counts all paths of length n, n>=0, starting at the initial node on the path graph P_7, see the Maple program. (End)

Range of row n of the circular Pascal array of order 8. - Shaun V. Ault, Jun 05 2014.

LINKS

Table of n, a(n) for n=0..31.

Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, arXiv:1407.2197 [math.CO], 2014.

Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, October 2014, Pages 45-54.

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Joseph Myers, BMO 2008-2009 Round 1 Problem 1 - Generalisation

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2)

FORMULA

a(0)=a(1)=1, a(2)=2, a(3)=3, a(n)=4*a(n-2)-2*a(n-4). - Harvey P. Dale, May 11 2011

a(n) = (2+sqrt(2+sqrt(2)))/8*(sqrt(2+sqrt(2)))^n + (2-sqrt(2+sqrt(2)))/8*(-sqrt(2+sqrt(2)))^n + (2+sqrt(2-sqrt(2)))/8*(sqrt(2-sqrt(2)))^n + (2-sqrt(2-sqrt(2)))/8*(-sqrt(2-sqrt(2)))^n. - Sergei N. Gladkovskii, Aug 23 2012

a(n) = A030435(n)/2. a(2*n) = A006012(n). a(2*n + 1) = A007022(n). - Michael Somos, Mar 06 2003

EXAMPLE

G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 34*x^7 + 68*x^8 + ...

MAPLE

with(GraphTheory): P:=7: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=31; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..P); od: seq(a(n), n=0..nmax); # Johannes W. Meijer, May 29 2010

MATHEMATICA

CoefficientList[Series[(1+x-2x^2-x^3)/(1-4x^2+2x^4), {x, 0, 40}], x] (* or *) LinearRecurrence[{0, 4, 0, -2}, {1, 1, 2, 3}, 41] (* Harvey P. Dale, May 11 2011 *)

PROG

(PARI) Vec((1+x-2*x^2-x^3)/(1-4*x^2+2*x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

(PARI) {a(n) = if( n<0, 0, polsym( x^4 - 4*x^2 + 2, n + n%2)[n + n%2 + 1] / (4 * (n%2 + 1)))}; /* Michael Somos, Feb 08 2015 */

CROSSREFS

Cf. A006012, A007022, A030435.

a(n) = A094718(7, n).

Cf. A024175, A094803, A000045, A038754, A028495, A030436, A061551 and A178381.

Sequence in context: A305889 A135452 A077027 * A030227 A180272 A319436

Adjacent sequences:  A030433 A030434 A030435 * A030437 A030438 A030439

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 11 1999

EXTENSIONS

Comment and link added and typo in cross-reference corrected by Joseph Myers, Dec 24 2008, May 30 2010

STATUS

approved

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Last modified November 21 03:20 EST 2018. Contains 317427 sequences. (Running on oeis4.)