1,2

This sequence is known to be a permutation of positive integers, along with its absolute difference sequence (A129199).

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).

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).

P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific J. Math., 71, 1977

(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)}

The absolute difference is in A129199, Cf. A081145, which can be called as Slater-Velez permutation sequence of the 1st kind.

Sequence in context: A091809 A110315 A094744 this_sequence A122442 A162613 A120858

Adjacent sequences: A129195 A129196 A129197 this_sequence A129199 A129200 A129201

nonn

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), Apr 04 2007