%I A005228 M2629
%S A005228 1,3,7,12,18,26,35,45,56,69,83,98,114,131,150,170,191,213,236,260,285,
%T A005228 312,340,369,399,430,462,495,529,565,602,640,679,719,760,802,845,889,935,
%U A005228 982,1030,1079,1129,1180,1232,1285,1339,1394,1451,1509,1568,1628,1689
%N A005228 Sequence and first differences (A030124) include all numbers exactly 
               once.
%C A005228 This is the lexicographically earliest sequence that together with its 
               first differences (A030124) contain every positive integer exactly 
               once.
%D A005228 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005228 E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, 
               Volume 59 (Jeux math'), April/June 2008, Paris.
%D A005228 D. Hofstadter, "Goedel, Escher, Bach", p. 73.
%D A005228 Clark Kimberling, Complementary Equations, Journal of Integer Sequences, 
               Vol. 10 (2007), Article 07.1.4.
%H A005228 T. D. Noe, <a href="b005228.txt">Table of n, a(n) for n=1..1000</a>
%H A005228 A. S. Fraenkel, <a href="http://www.integers-ejcnt.org/">New games related 
               to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial 
               Number Theory, Vol. 4, Paper G6, 2004.
%H A005228 Catalin Francu, <a href="a005228.txt">C++ program</a>
%H A005228 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A005228 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HofstadterFigure-FigureSequence.html">Link to a section of The World 
               of Mathematics.</a>
%H A005228 <a href="Sindx_Go.html#GEB">Index entries for sequences from "Goedel, 
               Escher, Bach"</a>
%H A005228 <a href="Sindx_Ho.html#Hofstadter">Index entries for Hofstadter-type 
               sequences</a>
%F A005228 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).
%F A005228 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)
%F A005228 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
%e A005228 Sequence reads 1 3 7 12 18 26 35 45...,
%e A005228 differences are 2 4 5, 6, 8, 9, 10 ... and
%e A005228 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!
%p A005228 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.
%t A005228 a = {1}; d = 2; k = 1; Do[ While[ Position[a, d] != {}, d++ ]; k = k 
               + d; d++; a = Append[a, k], {n, 1, 55} ]; a
%Y A005228 Cf. A030124 (complement), A056731, A056738, A061577, A037257, A140778.
%Y A005228 Sequence in context: A055998 A066379 A024517 this_sequence A000969 A122250 
               A024388
%Y A005228 Adjacent sequences: A005225 A005226 A005227 this_sequence A005229 A005230 
               A005231
%K A005228 nonn,easy,nice
%O A005228 1,2
%A A005228 N. J. A. Sloane (njas(AT)research.att.com).
%E A005228 Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 24 
               2001