A024431 - OEIS

A024431

A generalized difference set on the set of all integers (lambda = 1).

1, 2, 6, 8, 18, 21, 44, 52, 106, 115, 232, 243, 488, 502, 1006, 1024, 2050, 2071, 4144, 4166, 8334, 8358, 16718, 16743, 33488, 33515, 67032, 67060, 134122, 134151, 268304, 268334, 536670, 536702, 1073406, 1073439, 2146880, 2146915, 4293832

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OFFSET

0,2

COMMENTS

In the set of all positive differences of the sequence each integer appears exactly once, i.e., lambda = 1.

REFERENCES

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.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

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 not of form x-y for x, y in N_i, x>y. Union of all N_i gives sequence.

a(A115406(n)) - a(A115407(n)) = n; a(m) - a(n) = A115409(m*(m-1)/2+n+1), 1 <= n < m. - Reinhard Zumkeller, Jan 22 2006

For n > 0: a(n) = A115409(n,1) + 1. - Reinhard Zumkeller, Sep 16 2014

MAPLE

M:= 100: # to get all differences up to M

Agenda:= Array(1..M, 1):

a[1]:= 1: a[2]:= 2: Agenda[1]:= 0:

for n from 2 by 2 do

dm:= ArrayTools:-SearchArray(Agenda, 1);

if ArrayTools:-Size(dm)[1]=0 then break fi;

dm:= dm[1];

Agenda[dm]:= 0:

a[n+1]:= 2*a[n]+2:

a[n+2]:= a[n+1] + dm;

for j from n by -1 to 1 while a[n+1] - a[j] <= M do

Agenda[a[n+1]-a[j]]:= 0;

if a[n+2]-a[j] <= M then Agenda[a[n+2]-a[j]]:= 0 fi

od:

seq(a[i], i=1..n); # Robert Israel, Oct 08 2015

MATHEMATICA

differenceQ[seq_, x_] := (r = False; Do[ If[ x == seq[[k]] - seq[[j]], r = True; Break[]], { j, 1, Length[seq] }, {k, 1, Length[seq] } ]; r); seq[1] = {1, 2}; seq[i_] := seq[i] = (k = Max[ seq[i-1] ]; j = First[ Select[ Range[k], !differenceQ[ seq[i-1], #] & , 1]]; Union[ seq[i-1], {2k+2, 2k+2+j} ] ); A024431 = seq[20] (* Jean-François Alcover, Jan 04 2012 *)

PROG

(Haskell)

import Data.List ((\\))

a024431 n = a024431_list !! n

a024431_list = 1 : 2 : f [2, 1] [2 ..] where

f ks@(k:_) (j:js) =

x : y : f (y : x : ks) ((js \\ map (y -) ks) \\ map (x -) ks)

where y = x + j; x = 2 * k + 2

-- Reinhard Zumkeller, Sep 16 2014

CROSSREFS

Cf. A005282, A049399, A115408, A115409.

Cf. A247414 (first differences).