A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

1, 2, 4, 8, 16, 21, 42, 51, 102, 112, 224, 235, 470, 486, 972, 990, 1980, 2002, 4004, 4027, 8054, 8078, 16156, 16181, 32362, 32389, 64778, 64806, 129612, 129641, 259282, 259313, 518626, 518658, 1037316, 1037349, 2074698, 2074734, 4149468

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

This is a B_2 sequence. More economical recursion: a(1)=1, a(2n)=2a(2n-1), a(2n+1)=a(2n)+r(n), where r(n) is the smallest positive integer not of the form a(j)-a(i) with 1<=iA247556. - Thomas Ordowski, Sep 28 2014

References

• R. K. Guy, Unsolved Problems in Number Theory, E25.
• W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 444.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• R. L. Graham, Problem E1910, Amer. Math. Monthly, 73 (1966), 775.
• R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
• R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
• R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
• M. Hall, Cyclic projective planes, Duke Math. J., 4 (1947), 1079-1090.
• C. B. A. Peck, Remark on Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
• W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
• N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)

Crossrefs

Cf. A054540, A004978, A247556.

Programs

Mathematica

a[1] = 1; a[2] = 2; a[n_?OddQ] := a[n] = 2*a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + r[(n-2)/2]; r[n_] := ( diff = Table[a[j] - a[i], {i, 1, 2*n+1}, {j, i+1, 2*n+1}] // Flatten // Union; max = diff // Last; notDiff = Complement[Range[max], diff]; If[notDiff == {}, max+1, notDiff // First]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 31 2012 *)

Formula

a(1)=1, a(2)=2, a(2n+1) = 2a(2n), a(2n+2) = a(2n+1) + r(n), where r(n) = smallest positive number not of form a(j) - a(i) with 1 <= i < j <= 2n+1.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 14 2000