|
| |
|
|
A068911
|
|
Number of n step walks (each step +/-1 starting from 0) which are never more than 2 or less than -2.
|
|
13
|
|
|
|
1, 2, 4, 6, 12, 18, 36, 54, 108, 162, 324, 486, 972, 1458, 2916, 4374, 8748, 13122, 26244, 39366, 78732, 118098, 236196, 354294, 708588, 1062882, 2125764, 3188646, 6377292, 9565938, 19131876, 28697814, 57395628, 86093442, 172186884
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Contribution from Johannes W. Meijer, May 29 2010: (Start)
The a(n) represent the number of ways White can force checkmate in exactly (n+1) moves, n>=0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6, g5 and h6; Black Ke8, Nh8, pawns b3, c7, d3, f7, g6 and h7. (After Noam D. Elkies, see link; diagram 5).
Counts all paths of length n, n>=0, starting at the third node on the path graph P_5, see the Maple program.
(End)
|
|
|
LINKS
|
Table of n, a(n) for n=0..34.
Noam D. Elkies, New Directions in Enumerative Chess Problems., The Electronic Journal of Combinatorics, 11 (2), 2004-2005. [From Johannes W. Meijer, May 29 2010]
|
|
|
FORMULA
|
a(n) =A068913(2, n) =2*A038754(n-1) =3*a(n-2) =a(n-1)*a(n-2)/a(n-3) starting with a(0)=1, a(1)=2, a(2)=4 and a(3)=6. For n>0: a(2n)=4*3^(n-1)=2*a(2n-1); a(2n+1)=2*3^n=3*a(2n)/2=2*a(2n)-A000079(n-2).
G.f.: (1+x)^2/(1-3x^2); a(n)=2*3^((n+1)/2)((1-(-1)^n)/6+sqrt(3)(1+(-1)^n)/9)-0^n/3. The sequence 0, 1, 2, 4, ... has a(n)=2*3^(n/2)((1+(-1)^n)/6+sqrt(3)(1-(-1)^n)/9)-2*0^n/3+sum{k=0..n, binom(n, k)k(-1)^k}/3 - Paul Barry, Feb 17 2004
|
|
|
MAPLE
|
with(GraphTheory): G:= PathGraph(5): A:=AdjacencyMatrix(G): nmax:=34; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[3, k], k=1..5) od: seq(a(n), n=0..nmax); [From Johannes W. Meijer, May 29 2010]
|
|
|
MATHEMATICA
|
Join[{1}, Transpose[NestList[{Last[#], 3First[#]}&, {2, 4}, 40]][[1]]] (* From Harvey P. Dale, Oct 24 2011 *)
|
|
|
CROSSREFS
|
Cf. A000007, A016116 (without initial term), A068912, A068913 for similar.
A060647(n-1)+1.
Cf. A028495, A038754, A048328, A078038 and A124302.
Sequence in context: A063516 A104352 A133488 * A094769 A068018 A060798
Adjacent sequences: A068908 A068909 A068910 * A068912 A068913 A068914
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Henry Bottomley, Mar 06 2002
|
|
|
STATUS
|
approved
|
| |
|
|
