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Search: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114
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| A005228 |
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Sequence and first differences (A030124) include all numbers exactly once.
(Formerly M2629)
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+20
25
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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This is the lexicographically earliest sequence that together with its first differences (A030124) contain every positive integer exactly once.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
D. Hofstadter, "Goedel, Escher, Bach", p. 73.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for Hofstadter-type sequences
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FORMULA
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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
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EXAMPLE
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Sequence reads 1 3 7 12 18 26 35 45...,
differences are 2 4 5, 6, 8, 9, 10 ... and
the point is 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!
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MAPLE
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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.
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MATHEMATICA
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a = {1}; d = 2; k = 1; Do[ While[ Position[a, d] != {}, d++ ]; k = k + d; d++; a = Append[a, k], {n, 1, 55} ]; a
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CROSSREFS
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Cf. A030124 (complement), A056731, A056738, A061577, A037257, A140778.
Sequence in context: A055998 A066379 A024517 this_sequence A000969 A122250 A024388
Adjacent sequences: A005225 A005226 A005227 this_sequence A005229 A005230 A005231
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 24 2001
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| A035253 |
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Second differences are 2,2,1,2,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,1,2,.. (A035214). |
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+20
3
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| 4, 1, 0, 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, 286, 313, 341, 370, 400, 431, 463, 497, 532, 568, 605, 643, 682, 722, 763, 806, 850, 895, 941, 988, 1036, 1085, 1135
(list; graph; listen)
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