A054668 Number of distinct non-extendable sequences X={x(1),x(2),...,x(k)} where x(1)=1, the x(i)'s are distinct elements of {1,...,n} with |x(i)-x(i+1)|=1 or 2, for i=1,2,...,k.

1, 2, 4, 8, 16, 30, 56, 104, 192, 354, 652, 1200, 2208, 4062, 7472, 13744, 25280, 46498, 85524, 157304, 289328, 532158, 978792, 1800280, 3311232, 6090306, 11201820, 20603360, 37895488, 69700670, 128199520, 235795680, 433695872, 797691074, 1467182628, 2698569576

Offset

1,2

Comments

Superseeker suggests the g.f. ((x^4)+1)/(x^4-2x+1). If the sequences X, in the enumeration of a(n), are required to contain n then sequence A000073 (tribonacci numbers) is obtained.

Links

Table of n, a(n) for n=1..36.

Formula

a(1) = 1, a(n) = term (4,2) in the 4 X 4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,0; 2,0,0,1]^n (n > 1). - Alois P. Heinz, Jul 24 2008

Example

a(4) = 4, since the allowable sequences are {1,2,3,4}, {1,2,4,3}, {1,3,2,4} and {1,3,4,2}, whereas {1,4,2,3} and {1,4,3,2} violate the spacing condition.

Maple

a:= n-> `if`(n=1, 1, (<<1|1|0|0>, <1|0|1|0>, <1|0|0|0>, <2|0|0|1>>^n)[4, 2]):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 24 2008

Mathematica

a=b=c=0; Join[{1}, Table[d=a+b+c+2; a=b; b=c; c=d, {n, 50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

Crossrefs

Cf. A053623.

Sequence in context: A125548 A227298 A164229 * A164225 A164204 A164209

Adjacent sequences: A054665 A054666 A054667 * A054669 A054670 A054671

Keyword

nonn

Author

John W. Layman, Apr 18 2000

Status

approved