A115406 -id:A115406 - 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

A115408 Integer permutation induced by A024431.

Original entry on oeis.org

1, 6, 15, 3, 2, 5, 4, 28, 45, 10, 66, 9, 14, 91, 13, 8, 7, 120, 12, 11, 153, 190, 21, 231, 276, 20, 325, 378, 435, 496, 27, 561, 630, 26, 703, 19, 780, 18, 861, 946, 1035, 17, 16, 25, 1128, 24, 1225, 1326, 1431, 23, 22, 1540, 1653, 36, 1770, 1891, 2016, 2145

Offset: 1

Reinhard Zumkeller, Jan 22 2006

nonn

a(n) = A115406(n)*(A115406(n)-1)/2 + A115407(n) + 1;

inverse: A115409.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex); import Data.Maybe (fromJust)
a115408 = (+ 1) . fromJust . (`elemIndex` a115409_list)
-- Reinhard Zumkeller, Sep 16 2014

cp's OEIS Frontend

A115407 Unique m such that A024431(A115406(n)) - A024431(m) = n.

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

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A115408 Integer permutation induced by A024431.

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