A129198 Slater-Velez permutation sequence of the 2nd kind.

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

Offset: 1

Author

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

Keywords

• nonn

Comments

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).

Links

• 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.

Crossrefs

The absolute difference is in A129199.

Cf. A081145.

Programs

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)}