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A129198 Slater-Velez permutation sequence of the 2nd kind. 4
1, 2, 5, 3, 11, 7, 23, 4, 47, 42, 95, 89, 191, 6, 383, 376, 767, 8, 1535, 1526, 3071, 9, 6143, 6133, 12287, 10, 24575, 24564, 49151, 49139, 98303, 12, 196607, 196594, 393215, 13, 786431, 786417, 1572863, 14, 3145727, 3145712, 6291455, 15, 12582911 (list; graph; refs; listen; history; text; internal format)



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).


Table of n, a(n) for n=1..45.

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

William Y. Velez, Research problems 159-160, Discrete Math 110 (1992), pp. 301-302.


(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 first kind.

Sequence in context: A094744 A229608 A185061 * A122442 A225258 A162613

Adjacent sequences:  A129195 A129196 A129197 * A129199 A129200 A129201




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



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