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A005228
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Sequence and first differences (A030124) together include all positive numbers exactly once.
(Formerly M2629)
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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; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| This is the lexicographically earliest sequence that together with its first differences (A030124) contain every positive integer exactly once.
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REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Catalin Francu, C++ program
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
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
| 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
| 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
| 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
| 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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PROG
| (Haskell)
import Data.List (delete)
a005228 n = a005228_list !! (n-1)
a005228_list = 1 : figure 1 [2..] where
figure n (x:xs) = n' : figure n' (delete n' xs) where n' = n + x
-- Reinhard Zumkeller, Mar 03 2011
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CROSSREFS
| Cf. A030124 (complement), A056731, A056738, A061577, A037257, A140778.
Sequence in context: A055998 A066379 A024517 * A000969 A194117 A122250
Adjacent sequences: A005225 A005226 A005227 * A005229 A005230 A005231
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 24 2001
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