A049399 A generalized difference set on the set of all integers (lambda = 2).

1, 2, 6, 7, 16, 18, 38, 40, 82, 85, 172, 175, 352, 356, 714, 720, 1442, 1449, 2900, 2907, 5816, 5824, 11650, 11658, 23318, 23327, 46656, 46666, 93334, 93345, 186692, 186704, 373410, 373423, 746848, 746861, 1493724, 1493738, 2987478, 2987493, 5974988, 5975004

Offset: 0

Author

Otokar Grosek (grosek(AT)elf.stuba.sk)

Keywords

• nonn
• easy

Comments

In the set of all positive differences of the sequence each integer appears exactly twice, i.e., lambda = 2.

One could try to greedily build such a difference set as follows: b(1) = 1, b(n+1) = b(n)+j with j the smallest difference yet to appear twice. This would begin with {1, 2, 3, 5, 8, 12, 17, 23, 31, 39, 49} and fail; the smallest difference yet to appear twice is then 12 = 17-5, but 49+12 = 61 and 61-39 = 22 = 23-1 = 39-17. - Danny Rorabaugh, Sep 27 2015

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..2500
• T. Baginova, R. Jajcay, Notes on subtractive properties of natural numbers, Bulletin of the ICA, Vol. 25(1999), pp. 29-40
• O. Grosek, R. Jajcay, Generalized Difference Sets on an Infinite Cyclic Semigroup, JCMCC, Vol. 13 (1993), pp. 167-174.

Crossrefs

Cf. A024431.

Formula

Let N_1={1, 2}. Given N_i, let N_{i+1} = N_i union {2k+2, 2k+2+j} where k = max element of N_i and j = smallest number of form x-y for at most one pair x, y in N_i, x>y. Union of all N_i gives sequence. - Danny Rorabaugh (mirroring formula in A024431), Sep 27 2015

Extensions

a(12)-a(15) corrected and more terms added by Danny Rorabaugh, Sep 27 2015