A247414 -id:A247414 - cp's OEIS frontend

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

Offset: 0

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

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

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.

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

Cf. A005282, A049399, A115408, A115409.

Cf. A247414 (first differences).

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

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

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 *)

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

More terms from Larry Reeves (larryr(AT)acm.org), May 04 2000

A115409 Inverse integer permutation of A115408.

Original entry on oeis.org

1, 5, 4, 7, 6, 2, 17, 16, 12, 10, 20, 19, 15, 13, 3, 43, 42, 38, 36, 26, 23, 51, 50, 46, 44, 34, 31, 8, 105, 104, 100, 98, 88, 85, 62, 54, 114, 113, 109, 107, 97, 94, 71, 63, 9

Offset: 1

Reinhard Zumkeller, Jan 22 2006

Seen as a triangle read by rows T(n,k) = a(n*(n-1)/2+k) = A024431(n)-A024431(k-1), 1<=k<=n.

T(n,1) = A024431(n)-1; T(n,n) = A247414(n-1). - Reinhard Zumkeller, Sep 16 2014

			Triangle begins:
1;
5, 4;
7, 6, 2;
17, 16, 12, 10;
20, 19, 15, 13, 3;
...

Reinhard Zumkeller, >Rows n = 1..125 of triangle, flattened
Index entries for sequences that are permutations of the natural numbers

Cf. A024431, A115408, A247414.

import Data.List (inits)
a115409 n k = a115409_tabl !! (n-1) !! (k-1)
a115409_row n = a115409_tabl !! (n-1)
a115409_tabl = map f $ drop 2 $ inits a024431_list where
   f xs = reverse $ map (z -) zs where (z:zs) = reverse xs
a115409_list = concat a115409_tabl
-- Reinhard Zumkeller, Sep 16 2014

nmax = 9;
differenceQ[seq_, x_] := Module[{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] = Module[{j, k}, 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[nmax];
T[n_, k_] := A024431[[n+1]] -  A024431[[k]];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 20 2021 *)

cp's OEIS Frontend

A024431 A generalized difference set on the set of all integers (lambda = 1).

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