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A068911 Number of n step walks (each step +/-1 starting from 0) which are never more than 2 or less than -2. 18
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, 258280326, 516560652 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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)

From Alec Jones, Feb 25 2016: (Start)

The a(n) are the n-th terms in a "fibonacci-snake" drawn on a rectilinear grid. The n-th term is computed as the sum of the previous terms in cells adjacent to the n-th cell (diagonals included). (This sequence excludes the first term of the snake.) For example:

1 ...  1      1  ...   1 4      1 4 6 ...  1 4 6       1 4 6   ...  and so on.

       1 ...  1 2      1 2 ...  1 2        1 2 12 ...  1 2 12 18

(End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..4191

Robert Dorward et al., A Generalization of Zeckendorf's Theorem via Circumscribed m-gons, arXiv:1508.07531 [math.NT], 2015. See Example 1.3 p. 4.

Noam D. Elkies, New Directions in Enumerative Chess Problems, arXiv:math/0508645 [math.CO], 2005; The Electronic Journal of Combinatorics, 11 (2), 2004-2005.

D. Panario, M. Sahin, Q. Wang, W. Webb, General conditional recurrences, Applied Mathematics and Computation, Volume 243, Sep 15 2014, Pages 220-231.

Noriaki Sannomiya, H Katsura, Y Nakayama, Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion, arXiv preprint arXiv:1612.02285 [cond-mat.str-el], 2016-2017. See Table I, line 3.

Index entries for linear recurrences with constant coefficients, signature (0,3)

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) - (1/3)*0^n. The sequence 0, 1, 2, 4, ... has a(n) = 2*3^(n/2)*((1+(-1)^n)/6 + sqrt(3)*(1-(-1)^n)/9) - (2/3)*0^n + (1/3)*Sum_{k=0..n} binomial(n, k)*k*(-1)^k. - Paul Barry, Feb 17 2004

a(n) = 2^((3+(-1)^n)/2)*3^((2*n-3-(-1)^n)/4)-((1-(-1)^(2^n)))/6. - Luce ETIENNE, Aug 30 2014

For n > 2, indexing from 0, a(n) = a(n-1)+a(n-2) if n is odd, a(n-1)+a(n-2)+a(n-3) if n is even. - Alec Jones, Feb 25 2016

a(n) = 2*a(n-1) if n is even, a(n-1)+a(n-2) if n is odd. - Vincenzo Librandi, Feb 26 2016

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); # Johannes W. Meijer, May 29 2010

# second Maple program:

a:= proc(n) a(n):= `if`(n<2, n+1, (4-irem(n, 2))/2*a(n-1)) end:

seq(a(n), n=0..40);  # Alois P. Heinz, Feb 03 2019

MATHEMATICA

Join[{1}, Transpose[NestList[{Last[#], 3First[#]}&, {2, 4}, 40]][[1]]] (* Harvey P. Dale, Oct 24 2011 *)

PROG

(PARI) a(n)=[4, 6][n%2+1]*3^(n\2)\3 \\ Charles R Greathouse IV, Feb 26 2016

(MAGMA) [Floor((5-(-1)^n)*3^Floor(n/2)/3): n in [0..40]]; // Bruno Berselli, Feb 26 2016, after Charles R Greathouse IV.

CROSSREFS

Cf. A000007, A016116 (without initial term), A068912, A068913 for similar.

Equals A060647(n-1)+1.

Cf. also A028495, A038754, A048328, A078038, A124302, A306293.

Sequence in context: A306315 A104352 A133488 * A243543 A094769 A068018

Adjacent sequences:  A068908 A068909 A068910 * A068912 A068913 A068914

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Mar 06 2002

STATUS

approved

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Last modified December 11 18:19 EST 2019. Contains 329925 sequences. (Running on oeis4.)