A048162 - OEIS

%I #25 Oct 01 2023 12:20:39

%S 1,1,2,6,9,13,20,31,49,78,125,201,324,523,845,1366,2209,3573,5780,

%T 9351,15129,24478,39605,64081,103684,167763,271445,439206,710649,

%U 1149853,1860500,3010351,4870849,7881198,12752045,20633241,33385284

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

%C 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

%D 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.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -1).

%F For n>4, a(n) = a(n-1) + a(n-2) - 2. - _Joshua Zucker_, Nov 13 2003

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

%F For n > 2, a(n)=A001610(n-1) - 3. - _Toby Gottfried_, Apr 13 2013

%t 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 *)

%Y 3rd column of A008305.

%Y Cf. A001610.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Second formula corrected by _David Radcliffe_, Jan 16 2011