A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j. (Formerly M1112 N0425)

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

1,2

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<=i<j<=2n; gives the sequence: 1, 2, 4, 8, 13, 26, 34, ... which also grows exponentially. Effective optimization of this method, giving a sequence of polynomial growth, is proposed in A247556. - 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)

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.

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 *)

Crossrefs

Cf. A054540, A004978, A247556.

Sequence in context: A196871 A334423 A375006 * A328265 A002081 A102039

Adjacent sequences: A001853 A001854 A001855 * A001857 A001858 A001859

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved