A308021 -id:A308021 - cp's OEIS frontend

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.

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

Offset: 1

Don Reble, Mar 08 2003

nonn

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

			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.

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; Local copy
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.
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.
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

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.

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

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 *)

{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

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

A099004 First differences of A081145.

Original entry on oeis.org

1, 2, 3, -4, 5, 6, -9, 7, 8, -14, 10, 11, -18, 12, 13, -24, 15, 16, -30, 17, 19, -34, 20, 21, -39, 22, 23, -43, 25, 26, -50, 27, 28, -54, 29, 31, -57, 33, -32, 35, 36, -70, 37, 38, -73, 40, 41, -78, 42, 44, -85, 45, 46, -90, 47, 48, -94, 49, 51, -97, 52, 53, -104, 55, 56, -109, 58, 59, -116, 61, -60, 62, 63

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 17 2003

Does every number appear in the sequence of absolute values (see A308007)?

Chai Wah Wu, Table of n, a(n) for n = 1..5000

Cf. A081145, A081146, A308007, A308021.

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]]; t = Nest[f, {1}, 75]; Rest@t - Most@t (* Robert G. Wilson v, Jun 13 2006 *)

A099004_list, l, s, b1, b2 = [1], 2, 3, set(), set([1])
for n in range(2, 5*10**3+1):
    i = s
    while True:
        m = abs(i-l)
        if not (i in b1 or m in b2):
            A099004_list.append(i-l)
            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

Edited by N. J. A. Sloane, Apr 08 2006

More terms from Robert G. Wilson v, Jun 13 2006

A308007 Absolute values of first differences of A081145.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 7, 8, 14, 10, 11, 18, 12, 13, 24, 15, 16, 30, 17, 19, 34, 20, 21, 39, 22, 23, 43, 25, 26, 50, 27, 28, 54, 29, 31, 57, 33, 32, 35, 36, 70, 37, 38, 73, 40, 41, 78, 42, 44, 85, 45, 46, 90, 47, 48, 94, 49, 51, 97, 52, 53, 104, 55, 56, 109, 58, 59, 116, 61, 60, 62, 63

Offset: 1

N. J. A. Sloane, May 13 2019

nonn

The (signed) differences themselves are in A099004, but this sequence is important enough to have its own entry.
Conjectured (see the Slater-Velez and Velez articles) to be a permutation of the positive integers.
It appears that the terms line on two lines (this is true for the first million terms): see A308016-A308020.

Chai Wah Wu, Table of n, a(n) for n = 1..5000
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.
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.
William Y. Velez, Research problems 159-160, Discrete Math., 110 (1992), pp. 301-302.

Cf. A081145, A081146, A099004, A308021.

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A308049 Inverse permutation to A308021 (Recamán variant).

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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.

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A099004 First differences of A081145.

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A308007 Absolute values of first differences of A081145.

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