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A217694 Number of n-variations of the set {1,2,...,n+1} satisfying p(i)-i in {-2,0,2}, i=1..n (an n-variation of the set N_{n+s} = {1,2,...,n+s} is any 1-to-1 mapping p from the set N_n = {1,2,...,n} into N_{n+s} = {1,2,...,n+s}).

%I A217694 #28 Mar 25 2025 09:39:32
%S A217694 1,1,2,4,8,12,21,35,60,96,160,260,429,693,1134,1836,2992,4840,7865,
%T A217694 12727,20648,33408,54144,87608,141897,229593,371722,601460,973560,
%U A217694 1575252,2549421,4125051,6675460,10801120,17478176,28280284,45761045,74042925,119808150
%N A217694 Number of n-variations of the set {1,2,...,n+1} satisfying p(i)-i in {-2,0,2}, i=1..n (an n-variation of the set N_{n+s} = {1,2,...,n+s} is any 1-to-1 mapping p from the set N_n = {1,2,...,n} into N_{n+s} = {1,2,...,n+s}).
%H A217694 V. Baltic, <a href="https://doi.org/10.2298/YJOR120211023">Applications of the finite state automata for counting restricted permutations and variations</a>, Yugoslav Journal of Operations Research, 22 (2012), Number 2, 183-198. - _N. J. A. Sloane_, Jan 02 2013
%H A217694 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,2,-2,-1,-1,-1).
%F A217694 Recurrence: a(n)=a(n-1)+a(n-2)+2*a(n-4)-2*a(n-5)-a(n-6)-a(n-7)-a(n-8).
%F A217694 G.f.: (1+x^3)/(1-x-x^2-2*x^4+2*x^5+x^6+x^7+x^8) = (1+x)*(1-x+x^2)/((1-x-x^2)*(1+x^2)*(1-x^2-x^4)).
%t A217694 LinearRecurrence[{1,1,0,2,-2,-1,-1,-1},{1,1,2,4,8,12,21,35},40] (* _Harvey P. Dale_, Feb 29 2020 *)
%Y A217694 Cf. A072827, A072850-A072856, A079955-A080014.
%K A217694 nonn,easy
%O A217694 0,3
%A A217694 _Vladimir Baltic_, Oct 11 2012