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A081145 a(1)=1; thereafter, a(n) is the least positive integer which has not already occurred and is such that |a(n)-a(n-1)| is different from any |a(k)-a(k-1)| which has already occurred. 49
1, 2, 4, 7, 3, 8, 14, 5, 12, 20, 6, 16, 27, 9, 21, 34, 10, 25, 41, 11, 28, 47, 13, 33, 54, 15, 37, 60, 17, 42, 68, 18, 45, 73, 19, 48, 79, 22, 55, 23, 58, 94, 24, 61, 99, 26, 66, 107, 29, 71, 115, 30, 75, 121, 31, 78, 126, 32, 81, 132, 35, 87, 140, 36, 91, 147, 38, 96, 155, 39 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is a permutation of the positive integers. The inverse is A081146.

Similar to A100707, except that when we subtract we use the largest possible k.

The 1977 paper of Slater and Velez proves that this sequence is a permutation of positive integers and conjectures that its absolute difference sequence (see A308007) is also a permutation. If we call this the "Slater-Velez permutation of the first kind", then they also constructed another permutation (the 2nd kind), for which they are able to prove that both the sequence (A129198) and its absolute difference (A129199) are true permutations. - Ferenc Adorjan, Apr 03 2007

The points appear to lie on three straight lines of slopes roughly 0.56, 1.40, 2.24 (click "graph", or see the Wilks link). I checked this for the first 10^6 terms using Allan Wilks's C program. See A308009-A308015 for further information about the three lines. - N. J. A. Sloane, May 14 2019

REFERENCES

P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, II, Pacific Journal of Mathematics, Vol. 82, No. 2, 1979, 527-531.

LINKS

Ferenc Adorjan, Table of n,a(n) for n = 1..5000

Shalosh B. Ekhad and Doron Zeilberger, Guessing the Elusive Patterns in the Slater-Valez sequence (aka OEIS A081145), June 2019.

P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific Journal of Mathematics, Vol. 71, No. 1, 1977, 193-196.

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

Allan Wilks, Table showing n, a(n), slope, line_number, for n=1..100000 [The three lines are labeled 0 (lower), 1 (middle), 2 (upper).]

Allan Wilks, C program for A081145

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

a(4)=7 because the previous term is 4 and the differences |3-4|, |5-4| and |6-4| have already occurred.

After 7 we get 3 as the difference 4 has not occurred earlier. 5 follows 14 as the difference 9 has not occurred earlier.

MATHEMATICA

f[s_] := Block[{d = Abs[Rest@s - Most@s], k = 1}, While[ MemberQ[d, Abs[k - Last@s]] || MemberQ[s, k], k++ ]; Append[s, k]]; NestList[s, {1}, 70] (* Robert G. Wilson v, Jun 09 2006 *)

f[s_] := Block[{k = 1, d = Abs[Most@s - Rest@s], l = Last@s}, While[MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jun 13 2006 *)

PROG

(PARI){SV_p1(n)=local(x, v=6, d=2, j, k); /* Slater-Velez permutation - the first kind (by F. Adorjan)*/ x=vector(n); x[1]=1; x[2]=2; for(i=3, n, j=3; k=1; while(k, if(k=bittest(v, j)||bittest(d, abs(j-x[i-1])), j++, v+=2^j; d+=2^abs(j-x[i-1]); x[i]=j))); return(x)} \\ Ferenc Adorjan, Apr 03 2007

(Python)

A081145_list, l, s, b1, b2 = [1, 2], 2, 3, set(), set([1])

for n in range(3, 10**2):

....i = s

....while True:

........m = abs(i-l)

........if not (i in b1 or m in b2):

............A081145_list.append(i)

............b1.add(i)

............b2.add(m)

............l = i

............while s in b1:

................b1.remove(s)

................s += 1

............break

........i += 1 # Chai Wah Wu, Dec 15 2014

(Haskell)

import Data.List (delete)

a081145 n = a081145_list !! (n-1)

a081145_list = 1 : f 1 [2..] [] where

   f x vs ws = g vs where

     g (y:ys) = if z `elem` ws then g ys else y : f y (delete y vs) (z:ws)

                where z = abs (x - y)

-- Reinhard Zumkeller, Jul 02 2015

CROSSREFS

The sequence of differences is A099004 (see also A308007).

Similar to Murthy's sequence A093903, Cald's sequence (A006509) and Recamán's sequence A005132. See also A100707 (another version).

A308021 is an offspring of this sequence. - N. J. A. Sloane, May 13 2019

Cf. A063733, A072007, A078783, A081146, A084331, A084335, A117622.

See A308009-A308015 for the lines that the points lie on.

A308172 gives smallest missing numbers.

Sequence in context: A139696 A308049 A084332 * A100707 A302663 A078943

Adjacent sequences:  A081142 A081143 A081144 * A081146 A081147 A081148

KEYWORD

nonn,changed

AUTHOR

Don Reble, Mar 08 2003

STATUS

approved

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Last modified June 20 00:47 EDT 2019. Contains 324223 sequences. (Running on oeis4.)