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
Examples
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!
References
- 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).
Links
- 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
Crossrefs
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. A225850, A232746, A232747 (inverse), A232739, A232740, A232750 and also permutation pair A232751/A232752 constructed from this sequence and its complement.
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-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.
Programs
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Haskell
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 -
Maple
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
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Mathematica
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. *) -
PARI
A005228(n,print_all=0,s=1,used=0)={while(n--,used += 1<
Formula
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
Extensions
Additional comments from Robert G. Wilson v, Oct 24 2001
Incorrect formula removed by Benoit Jubin, May 13 2015
Comments