A099004 -id:A099004 - cp's OEIS frontend

A005228 Sequence and first differences (A030124) together list all positive numbers exactly once.

1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, 285, 312, 340, 369, 399, 430, 462, 495, 529, 565, 602, 640, 679, 719, 760, 802, 845, 889, 935, 982, 1030, 1079, 1129, 1180, 1232, 1285, 1339, 1394, 1451, 1509, 1568, 1628, 1689

Offset: 1

N. J. A. Sloane

This is the lexicographically earliest sequence that together with its first differences (A030124) contains every positive integer exactly once.
Hofstadter introduces this sequence in his discussion of Scott Kim's "FIGURE-FIGURE" drawing. - N. J. A. Sloane, May 25 2013
A225850(a(n)) = 2*n-1, cf. A167151. - Reinhard Zumkeller, May 17 2013
In view of the definition of A075326: start with a(0) = 0, and extend by rule that the next term is the sum of the predecessor and the most recent non-member of the sequence. - Reinhard Zumkeller, Oct 26 2014

			Sequence reads 1 3 7 12 18 26 35 45..., differences are 2 4 5, 6, 8, 9, 10 ... and the point is that every number not in the sequence itself appears among the differences. This property (together with the fact that both the sequence and the sequence of first differences are increasing) defines the sequence!

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 73.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10001 [The first 1000 terms were computed by T. D. Noe]
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Catalin Francu, C++ program
Cristian Francu, C program to generate the N-th element in O(sqrt(N))
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
Benoit Jubin, Asymptotic series for Hofstadter's figure-figure sequences, arXiv:1404.1791; J. Integer Sequences, 17 (2014), #14.7.2.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Eric Weisstein's World of Mathematics, Hofstadter Figure-Figure Sequence.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for Hofstadter-type sequences

Cf. A030124 (complement), A037257, A056731, A056738, A140778, A225687.
The following are a group of related sequences: A005132, A006509, A037257, A037258, A037259, A081145, A093903, A099004, A100707, A129198, A129199, A140778, A225376, A225377, A225378, A225385, A225386, A225387.
Cf. A075326, A095115.
Cf. A225850, A232746, A232747 (inverse), A232739, A232740, A232750 and also permutation pair A232751/A232752 constructed from this sequence and its complement.
Cf. A001651 (analog with sums instead of differences), A121229 (analog with products).
The same recurrence a(n) = a(n-1) + c(n-1) with different starting conditions: A061577 (starting with 2), A022935 (3), A022936 (4), A022937 (5), A022938 (6).
Related recurrences:
a(n-1) + c(n+1) - A022953, A022954.
a(n-1) + c(n) - A022946 to A022952.
a(n-1) + c(n-2) - A022940, A022941.
a(n-2) + c(n-1) - A022942 to A022944.
a(n-2) + c(n-2) - A022939.
a(n-3) + c(n-3) - A022955.
a(n-4) + c(n-4) - A022956.
a(n-5) + c(n-5) - A022957.

a005228 = scanl (+) 1 a030124
a030124 = go 1 a005228 where go x ys | x < head ys = x     : go (x + 1) ys
                                     | otherwise   = x + 1 : go (x + 2) (tail ys)
-- Maks Verver, Jun 30 2025

maxn := 5000; h := array(1..5000); h[1] := 1; a := [1]; i := 1; b := []; for n from 2 to 1000 do if h[n] <> 1 then b := [op(b), n]; j := a[i]+n; if j < maxn then a := [op(a),j]; h[j] := 1; i := i+1; fi; fi; od: a; b; # a is A005228, b is A030124.
A030124 := proc(n)
    option remember;
    local a,fnd,t ;
    if n <= 1 then
        op(n+1,[2,4]) ;
    else
        for a from procname(n-1)+1 do
            fnd := false;
            for t from 1 to n+1 do
                if A005228(t)  = a then
                    fnd := true;
                    break;
                end if;
            end do:
            if not fnd then
                return a;
            end if;
        end do:
    end if;
end proc:
A005228 := proc(n)
    option remember;
    if n <= 2 then
        op(n,[1,3]) ;
    else
        procname(n-1)+A030124(n-2) ;
    end if;
end proc: # R. J. Mathar, May 19 2013

a = {1}; d = 2; k = 1; Do[ While[ Position[a, d] != {}, d++ ]; k = k + d; d++; a = Append[a, k], {n, 1, 55} ]; a
(* Second program: *)
(* Program from Larry Morris, Jan 19 2017: *)
d = 3; a = {1, 3, 7, 12, 18}; While[ Length[a = Join[a, a[[-1]] + Accumulate[Range[a[[d]] + 1, a[[++d]] - 1]]]] < 50]; a
(* Comment: This adds as many terms to the sequence as there are numbers in each set of sequential differences. Consequently, the list of numbers it produces may be longer than the limit provided. With the limit of 50 shown, the sequence produced has length 60. *)

A005228(n,print_all=0,s=1,used=0)={while(n--,used += 1<M. F. Hasler, Feb 05 2013

a(n) = a(n-1) + c(n-1) for n >= 2, where a(1)=1, a( ) increasing, c( ) = complement of a( ) (c is the sequence A030124).
Let a(n) = this sequence, b(n) = A030124 prefixed by 0. Then b(n) = mex{ a(i), b(i) : 0 <= i < n}, a(n) = a(n-1) + b(n) + 1. (Fraenkel)
a(1) = 1, a(2) = 3; a( ) increasing; for n >= 3, if a(q) = a(n-1)-a(n-2)+1 for some q < n then a(n) = a(n-1) + (a(n-1)-a(n-2)+2), otherwise a(n) = a(n-1) + (a(n-1)-a(n-2)+1). - Albert Neumueller (albert.neu(AT)gmail.com), Jul 29 2006
a(n) = n^2/2 + n^(3/2)/(3*sqrt(2)) + O(n^(5/4)) [proved in Jubin link]. - Benoit Jubin, May 13 2015
For all n >= 1, A232746(a(n)) = n and A232747(a(n)) = n. [Both sequences work as left inverses of this sequence.] - Antti Karttunen, May 14 2015

Additional comments from Robert G. Wilson v, Oct 24 2001

Incorrect formula removed by Benoit Jubin, May 13 2015

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.

Original entry on oeis.org

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

A030124 Complement (and also first differences) of Hofstadter's sequence A005228.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Eric W. Weisstein

nonn

For any n, all integers k satisfying sum(i=1,n,a(i))+1Benoit Cloitre, Apr 01 2002
The asymptotic equivalence a(n) ~ n follows from the fact that the values disallowed in the present sequence because they occur in A005228 are negligible, since A005228 grows much faster than A030124. The next-to-leading term in the formula is calculated from the functional equation F(x) + G(x) = x, suggested by D. Wilson (cf. reference), where F and G are the inverse functions of smooth, increasing approximations f and f' of A005228 and A030124. It seems that higher order corrections calculated from this equation do not agree with the real behavior of a(n). - M. F. Hasler, Jun 04 2008
A225850(a(n)) = 2*n, cf. A167151. - Reinhard Zumkeller, May 17 2013

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
D. R. Hofstadter, "Gödel, Escher, Bach: An Eternal Golden Braid", Basic Books, 1st & 20th anniv. edition (1979 & 1999), p. 73.

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n=1..10000
Benoit Jubin, Asymptotic series for Hofstadter's figure-figure sequences, arXiv:1404.1791; J. Integer Sequences, 17 (2014), #14.7.2.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Eric Weisstein's World of Mathematics, Hofstadter Figure-Figure Sequence.
D. W. Wilson, Asymptotics about A005228, post to the SeqFan mailing list (access restricted to subscribers), Jun 03 2008
Index entries for sequences from "Goedel, Escher, Bach"

Cf. A005228, A030124, A037257, A037258, A037259, A061577, A140778, A129198, A129199, A100707, A093903, A005132, A006509, A081145, A099004, A225376, A225377, A225378, A225385, A225386, A225387, A225687.

import Data.List (delete)
a030124 n = a030124_list !! n
a030124_list = figureDiff 1 [2..] where
   figureDiff n (x:xs) = x : figureDiff n' (delete n' xs) where n' = n + x
-- Reinhard Zumkeller, Mar 03 2011

(* h stands for Hofstadter's sequence A005228 *) h[1] = 1; h[2] = 3; h[n_] := h[n] = 2*h[n-1] - h[n-2] + If[ MemberQ[ Array[h, n-1], h[n-1] - h[n-2] + 1], 2, 1]; Differences[ Array[h, 69]] (* Jean-François Alcover, Oct 06 2011 *)

{a=b=t=1;for(i=1,100, while(bittest(t,b++),); print1(b",");t+=1<M. F. Hasler, Jun 04 2008

a(n) = n + sqrt(2n) + o(n^(1/2)). - M. F. Hasler, Jun 04 2008 [proved in Jubin's paper].

Changed offset to agree with that of A005228. - N. J. A. Sloane, May 19 2013

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.

A308021 Start at n=1. Fill in a(n) with a value d > 0 not used earlier such that n-d or n+d is the smallest possible index not visited earlier, then continue with that index.

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 4, 6, 12, 15, 17, 8, 20, 9, 22, 11, 25, 27, 29, 14, 13, 33, 35, 37, 16, 40, 18, 19, 42, 45, 47, 49, 21, 24, 52, 55, 23, 58, 61, 62, 30, 26, 65, 66, 28, 68, 34, 31, 71, 74, 76, 79, 81, 39, 32, 83, 86, 36, 89, 43, 38, 91, 93, 96, 99, 41, 101, 50, 103, 106, 44, 108, 54, 111, 46, 113, 117, 48, 57, 118, 51, 120, 123

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 09 2019

nonn

A variant of Recamán's sequence: start at n=1, a(1)=1, then iterate: let n' =  n - a(n) if this n' > 0 and was not visited earlier, otherwise n' = n + a(n). Then let n'' <> n' be the smallest index not visited earlier (a(n'') not yet defined) such that the value |n'-n''| was not yet used (meaning not yet a value of any a(i)). Set a(n') = |n'-n''| and continue with n = n'. [Name and Comment suggested by M. F. Hasler at the request of the authors.]
Conjectured to be a permutation of the positive integers. The conjectured inverse permutation is given in A308049. - M. F. Hasler, May 10 2019
From Rémy Sigrist and N. J. A. Sloane, May 13 2019: (Start)
The sequence is given by the following formula. Let R(t) = A081145(t). Then for all t >= 1, a(R(t)) = |R(t+1)-R(t)|.
For example, for t=10, R(10)=20, R(11)=6, and a(R(10)) = a(20) = |6-20| = 14.
Since it is known that {R(t): t>=1} is a permutation of the positive integers (it is the "Slater-Velez permutation of the first kind"), this specifies a(n) for all n.
The connection with the definition as interpreted above by M. F. Hasler is that at step t of the procedure, n' is R(t) = A081145(t), n'' is R(t+1) = A081145(t+1), and we calculate a(R(t)) = |n'-n''| = |R(t)-R(t+1)|.
The conjecture that {a(n)} is a permutation of the positive integers is equivalent to Slater and Velez's conjecture (see references) that the absolute values of the first differences of A081145 are also a permutation of the positive integers. This problem appears to be still unsolved. (End)

			a(1) = 1 drives us to the empty cell a(2) since we can't go further to the left. We fill this cell with the number 2 which is the smallest integer not used before and thus allows us to go to the leftmost possible empty cell, 2 + 2 = 4. (There are no empty places to the left and we can't go to 3 = 2 + 1 since a step 1 has already been used.) So we have a(2) = 2.
a(2) = 2 drives us to the empty cell a(4). We see that the leftmost empty cell a(3) cannot be reached from a(4) since a step of 1 has already been used. We thus fill the cell a(4) with the smallest integer not used before, a(4) = 3.
a(4) = 3 drives us to the empty cell a(7). We see that the leftmost empty cell a(3) can now be reached from a(7) if we fill a(7) with 4; we have thus a(7) = 4.
a(7) = 4 drives us to the empty cell a(3), which is the one we wanted to fill. We fill a(3) with 5 which is the smallest integer not leading to a contradiction, whence a(3) = 5.
a(3) = 5 drives us to the empty cell a(8). We would like to fill this cell with 3, as this 3 would allow us to fill the leftmost empty cell of the sequence - but 3 has been used before; thus we'll have a(8) = 6.
a(8) = 6 drives us to the empty cell a(14). We fill a(14) with 9 as this will allow us to reach the leftmost empty cell of the sequence, whence a(14) = 9.
a(14) = 9 drives us to the empty cell a(5). We fill a(5) with 7 as this is the smallest integer not leading to a contradiction, so we have a(5) = 7, etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..5564
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. A005132 (Recamán's sequence), A171884 (injective variant).
Cf. A308049 (conjectured inverse permutation).
See also A081145, A081146, A099004.

{A=vector(N=199); n=1; while (n<=N,S=Set(A); Z=select(t->!t,A,1); for (i=1,#Z,Z[i]!=n||next; setsearch(S, abs(n-z=Z[i]))&& next; A[n]=abs(n-z); n=z; next(2)); break); if(#Z, A[1..Z[1]-!A[Z[1]]], A)} \\ M. F. Hasler, May 09 2019

A235263 a(n) = A235262(n) XOR A235262(n+1), where XOR stands for the binary XOR operator.

Original entry on oeis.org

3, 1, 7, 2, 14, 13, 12, 5, 11, 10, 6, 4, 31, 26, 27, 9, 22, 28, 8, 15, 53, 51, 50, 55, 52, 54, 48, 61, 58, 56, 62, 57, 59, 23, 39, 49, 20, 47, 44, 46, 43, 42, 45, 25, 29, 30, 17, 16, 18, 21, 19, 124, 107, 106, 109, 110, 111, 24, 118, 117, 114, 125, 127, 104

Offset: 1

Paul Tek, Jan 05 2014

nonn

All terms are distinct.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

Cf. A099004, A235262.

Perl
```
See Link section.
```

A258137 Absolute first differences of the lexicographically earliest sequence of odd positive integers such that the terms and their absolute first differences are all distinct.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 18, 14, 16, 28, 20, 22, 36, 24, 26, 48, 30, 32, 60, 34, 38, 68, 40, 42, 78, 44, 46, 86, 50, 52, 100, 54, 56, 108, 58, 62, 114, 66, 64, 70, 72, 140, 74, 76, 146, 80, 82, 156, 84, 88, 170, 90, 92, 180, 94, 96, 188, 98, 102, 194, 104, 106, 208

Offset: 1

Eric Angelini and Alois P. Heinz, May 21 2015

nonn

All terms are even.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
E. Angelini, Derangements on the SeqFan list, May 21 2015

Absolute first differences of A258136.

Cf. A099004, A257941, A257944.

b:= proc() false end:
g:= proc(n) option remember; local k;
      if n=1 then b(1):= true; 1
    else g(n-1); for k while b(k) or
         b(abs(g(n-1)-k)) by 2 do od;
         b(k), b(abs(g(n-1)-k)):= true$2; k
      fi
    end:
a:= n-> abs(g(n+1)-g(n)):
seq(a(n), n=1..101);

b[_] = False;
g[n_] := g[n] = Module[{k},
     If[n == 1, b[1] = True; 1,
     g[n-1]; For[k = 1, b[k] ||
     b[Abs[g[n-1] - k]], k += 2];
     {b[k], b[Abs[g[n-1] - k]]} = {True, True}; k]];
a[n_] := Abs[g[n+1] - g[n]];
Table[a[n], {n, 1, 101}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

a(n) = abs(A258136(n+1)-A258136(n)).

a(n) = 2*abs(A099004(n)).

A371295 Value k at the n-th step of A371282, where multiplied values are positive and subtracted values are negative.

Original entry on oeis.org

2, 3, -1, 4, -17, 5, -11, 6, -16, 7, -49, 9, -54, 8, -62, 10, -89, 12, -120, 13, -143, 14, -168, 15, -194, 18, -271, 19, -305, 20, -341, 21, -378, 22, -440, 23, -483, 24, -527, 25, -599, 26, -649, 27, -701, 28, -755, 29, -811, 30, -869, 31, -929, 32, -991, 33

Offset: 1

Neal Gersh Tolunsky, Mar 17 2024

sign

			At the first step of A371282, to get from A371282(1)=1 to A371282(2)=2, we multiply by 2, so a(1)=2.
At the fifth step of A371282, to get from A371282(5)=20 to A371282(6)=3, we subtract 17, so a(5)=-17.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A371282, A099004 (steps of add or subtract), A360597 (steps of multiply or divide).

from itertools import islice
def agen(): # generator of terms
    mina, an, aset, mink, kset = 1, 1, {1}, 1, set()
    while True:
        k1, ak1, k2 = 0, mina, mink
        if mina < an:
            for ak1 in range(mina, an-mink+1):
                if ak1 not in aset and an - ak1 not in kset:
                    k1 = an - ak1
                    break
        while k2 in kset or an*k2 in aset:
            k2 += 1
        an, k = (an-k1, k1) if k1 > 0 else (an*k2, k2)
        yield -k if k1 > 0 else k
        aset.add(an)
        kset.add(k)
        while mina in aset: mina += 1
        while mink in kset: mink += 1
print(list(islice(agen(), 56))) # Michael S. Branicky, Mar 18 2024

a(12) and beyond from Michael S. Branicky, Mar 18 2024

cp's OEIS Frontend

A005228 Sequence and first differences (A030124) together list all positive numbers exactly once.

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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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A030124 Complement (and also first differences) of Hofstadter's sequence A005228.

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A308007 Absolute values of first differences of A081145.

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A308021 Start at n=1. Fill in a(n) with a value d > 0 not used earlier such that n-d or n+d is the smallest possible index not visited earlier, then continue with that index.

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A235263 a(n) = A235262(n) XOR A235262(n+1), where XOR stands for the binary XOR operator.

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A258137 Absolute first differences of the lexicographically earliest sequence of odd positive integers such that the terms and their absolute first differences are all distinct.

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