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
Keywords
Examples
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.
Links
- Elisa Heinrich Mora and Noah A. Rosenberg, An nth-cousin mating model and the n-anacci numbers, arXiv:2506.16577 [q-bio.PE], 2025. See p. 7.
Crossrefs
Cf. A053623.
Programs
-
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 *)
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
Comments