A007086 Next term is uniquely the sum of 3 earlier terms.

1, 2, 3, 6, 9, 10, 11, 12, 28, 29, 30, 53, 56, 57, 80, 82, 104, 105, 107, 129, 130, 132, 154, 155, 157, 179, 180, 182, 204, 205, 207, 229, 230, 232, 254, 255, 257, 279, 280, 282, 304, 305, 307, 329, 330, 332, 354, 355, 357, 379, 380, 382, 404, 405, 407, 429

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn

Comments

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

Examples

			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_
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• 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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992

Crossrefs

Cf. A002858.

Programs

Mathematica

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 *)

Formula

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