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%I #18 Dec 05 2022 10:15:51
%S 1,2,5,3,11,7,23,4,47,42,95,89,191,6,383,376,767,8,1535,1526,3071,9,
%T 6143,6133,12287,10,24575,24564,49151,49139,98303,12,196607,196594,
%U 393215,13,786431,786417,1572863,14,3145727,3145712,6291455,15,12582911
%N Slater-Velez permutation sequence of the 2nd kind.
%C This sequence is known to be a permutation of positive integers, along with its absolute difference sequence (A129199).
%C The rule for constructing the sequence is as follows: a(1)=1, a(2)=2, then apply the following recursion, assuming the members are already present up to index k: let M(k)=max(a(1),a(2),...,a(k)) and let n(k) be the smallest positive integer not present in the sequence yet, while m(k) the smallest integer not present in the absolute difference sequence (d(1),d(2),...,d(k-1)), so far. Then a(k+1)=2M(k)+1 and if m(k)<=n(k) then set a(k+2)=a(k+1)-m(k), else a(k+2)=n(k).
%C In the paper of Slater and Velez it is shown that both the sequence a(n) and d(n) are permutations of positive integers (in spite of their strange appearance).
%H P. J. Slater and W. Y. Velez, <a href="https://projecteuclid.org/euclid.pjm/1102811644">Permutations of the Positive Integers with Restrictions on the Sequence of Differences</a>, Pacific J. Math., 71, 1977.
%H William Y. Velez, <a href="https://doi.org/10.1016/0012-365X(92)90724-T">Research problems 159-160</a>, Discrete Math 110 (1992), pp. 301-302.
%o (PARI) {SV_p2(n)=local(x,v=6,d=2,j,k,mx=2,nx=3,nd=2,u,w); /* Slater-Velez permutation - the 2nd kind */ x=vector(n);x[1]=1;x[2]=2; forstep(i=3,n,2,k=x[i]=2*mx+1;if(nd<=nx,j=x[i]-nd,j=nx);x[i+1]=j;mx=max(mx,max(j,k));v+=2^k+2^j; u=abs(k-x[i-1]);w=abs(j-k);d+=2^u+2^w;print(i" "k" "j" "u" "w); while(bittest(v,nx),nx++);while(bittest(d,nd),nd++)); return(x)}
%Y The absolute difference is in A129199.
%Y Cf. A081145.
%K nonn
%O 1,2
%A Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), Apr 04 2007
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