A129199 -id:A129199 - 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

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

Original entry on oeis.org

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

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

nonn

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).

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.

The absolute difference is in A129199.

Cf. A081145.

{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)}

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

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