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%I M0756 #26 Nov 07 2017 18:20:28
%S 1,2,3,6,9,10,11,12,28,29,30,53,56,57,80,82,104,105,107,129,130,132,
%T 154,155,157,179,180,182,204,205,207,229,230,232,254,255,257,279,280,
%U 282,304,305,307,329,330,332,354,355,357,379,380,382,404,405,407,429
%N Next term is uniquely the sum of 3 earlier terms.
%C a(1)=1, a(2)=2, a(3)=3, for n>3, a(n) = least number which is a unique sum of three distinct earlier terms. Written this way, we see that this is to 3 as Ulam number A002858 is to 2. - _Jonathan Vos Post_
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D M. C. Wunderlich, The improbable behavior of Ulam's summation sequence, pp. 249-257 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
%H Alois P. Heinz, <a href="/A007086/b007086.txt">Table of n, a(n) for n = 1..1000</a>
%H R. G. Wilson, V, <a href="/A002858/a002858_3.pdf">Letter to N. J. A. Sloane, Sep. 1992</a>
%F G.f.: (22*x^18 -21*x^17 +x^16 -2*x^13 -7*x^12 -15*x^9 +2*x^8 +2*x^7 -2*x^5 -2*x^4 -x^3 -x^2 -x) / (-x^4+x^3+x-1). Conjectured and verified for n<=1100 - _Alois P. Heinz_, Jan 04 2011
%e 13 through 27 are not in the sequence because of nonuniqueness: 1+3+9=1+2+10=13, 1+3+10=2+3+9=14, 1+2+12=2+3+10=15, 1+6+9=2+3+11=16, 1+7+9=2+6+9=17, 3+6+9=1+6+11=18, 1+6+12=2+6+11=19, 1+9+10=2+6+12=20, 1+9+11=2+9+10=21, 1+10+11=2+9+11=22, 2+9+12=3+9+11=23, 1+11+12=3+9+12=24, 3+10+12=6+9+10=25, 3+11+12=6+9+11=26, 6+9+12=6+10+11=27. - _Jonathan Vos Post_
%t Clear[a]; a[n_ /; n <= 3] := n; a[n_] := a[n] = (t = Table[a[i]+a[j]+a[k], {i, 1, n-3}, {j, i+1, n-2}, {k, j+1, n-1}] // Flatten; Complement[Select[t // Tally, #[[2]] == 1&][[All, 1]], Array[a, n-1]] // Sort // First); Array[a, 56] (* _Jean-François Alcover_, Mar 11 2014 *)
%Y Cf. A002858.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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