A048162 Expansion of (1 - x + 3*x^3 - 2*x^4 - 3*x^5)/(1 - 2*x + x^3).

1, 1, 2, 6, 9, 13, 20, 31, 49, 78, 125, 201, 324, 523, 845, 1366, 2209, 3573, 5780, 9351, 15129, 24478, 39605, 64081, 103684, 167763, 271445, 439206, 710649, 1149853, 1860500, 3010351, 4870849, 7881198, 12752045, 20633241, 33385284

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of permutations of 1..n such that each position is fixed or moves to an adjacent position (with n considered adjacent to 1). For example, a(4) = 9 because there is the identity; 2 cyclic permutations; 4 swaps of one pair of adjacent entries; and 2 swaps of two pairs of adjacent entries. - Joshua Zucker, Nov 13 2003

References

• Lehmer, D. H.; Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Links

• Index entries for linear recurrences with constant coefficients, signature (2, 0, -1).

Crossrefs

3rd column of A008305.

Cf. A001610.

Programs

Mathematica

CoefficientList[Series[(1-x+3x^3-2x^4-3x^5)/(1-2x+x^3),{x,0,40}],x] (* or *) Join[{1,1,2},#[[3]]+#[[1]]+2&/@Partition[Fibonacci[Range[2,50]],3,1]] (* Harvey P. Dale, Apr 06 2017 *)

Formula

For n>4, a(n) = a(n-1) + a(n-2) - 2. - Joshua Zucker, Nov 13 2003

a(n) = Fibonacci(n+1) + Fibonacci(n-1) + 2, for n>2. - Jessa Lee (jessal(AT)comcast.net), Nov 25 2003

For n > 2, a(n)=A001610(n-1) - 3. - Toby Gottfried, Apr 13 2013

Extensions

Second formula corrected by David Radcliffe, Jan 16 2011