A131389 -id:A131389 - cp's OEIS frontend

A131391 Positions of positive integers in A131389, minus 1.

1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 21, 24, 25, 28, 29, 32, 33, 35, 37, 39, 41, 44, 46, 47, 49, 51, 53, 56, 57, 59, 61, 63, 65, 68, 69, 72, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 95, 97, 99, 101, 103, 105, 108, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 1

Clark Kimberling, Jul 06 2007

nonn

If A131389 is a permutation of the integers, as conjectured, then A131391 is the complement of A131392.

			The 5th positive integer to occur in A131391 is 6, which occurs in position 9, so that a(5) = 9 - 1 = 8.

Cf. A131388, A131389, A131390, A131392, A131393, A131394, A131395, A131396, A131397.

a(n) = -1 + position of the n-th positive integer in the sequence A131389=(0,1,2,-1,3,4,-2,-3,6,-4,5,...)

A131392 Positions of negative integers in A131389, minus 1.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 15, 18, 20, 22, 23, 26, 27, 30, 31, 34, 36, 38, 40, 42, 43, 45, 48, 50, 52, 54, 55, 58, 60, 62, 64, 66, 67, 70, 71, 74, 76, 78, 80, 82, 84, 86, 87, 89, 91, 93, 96, 98, 100, 102, 104, 106, 107, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 131, 132

Offset: 1

Clark Kimberling, Jul 06 2007

nonn

If A131389 is a permutation of the integers, as conjectured, then A131392 is the complement of A131391.

			The 5th negative integer to occur in A131391 is -5, which occurs in position 13, so that a(5)=13-1=12.

Cf. A131388, A131389, A131390, A131391, A131393, A131394, A131395, A131396, A131397.

a(n) = -1 + position of the n-th negative integer in the sequence A131389=(0,1,2,-1,3,4,-2,-3,6,-4,5,...)

A175007 Positions of nonnegative terms in A131389.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 12, 14, 17, 18, 20, 22, 25, 26, 29, 30, 33, 34, 36, 38, 40, 42, 45, 47, 48, 50, 52, 54, 57, 58, 60, 62, 64, 66, 69, 70, 73, 74, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 96, 98, 100, 102, 104, 106, 109, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128

Offset: 1

Clark Kimberling, Apr 03 2010

nonn

Is a(n+1)-a(n) always 1,2 or 3?

			The first 20 terms of A131389 are
0,1,2,-1,3,4,-2,-3,6,-4,5,7,-5,8,-6,-7,9,10,-8,11.
Positions occupied by nonnegatives are
1,2,3,5,6,9,11,12,14,17,18,20.

Cf. A131389, A175008.

A175008 Positions of negative terms in A131389.

Original entry on oeis.org

4, 7, 8, 10, 13, 15, 16, 19, 21, 23, 24, 27, 28, 31, 32, 35, 37, 39, 41, 43, 44, 46, 49, 51, 53, 55, 56, 59, 61, 63, 65, 67, 68, 71, 72, 75, 77, 79, 81, 83, 85, 87, 88, 90, 92, 94, 97, 99, 101, 103, 105, 107, 108, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 132, 133

Offset: 1

Clark Kimberling, Apr 03 2010

nonn

Is a(n+1)-a(n) always 1,2 or 3?

			The first 20 terms of A131389 are
0,1,2,-1,3,4,-2,-3,6,-4,5,7,-5,8,-6,-7,9,10,-8,11.
Positions occupied by negatives are
4,7,8,10,13,15,16,19.

Cf. A131389, A175007.

A175498 a(1)=1. a(n) = the smallest positive integer not occurring earlier such that a(n)-a(n-1) doesn't equal a(k)-a(k-1) for any k with 2 <= k <= n-1.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 5, 11, 7, 12, 9, 16, 8, 17, 15, 23, 13, 24, 18, 28, 14, 26, 19, 32, 20, 34, 21, 36, 25, 41, 22, 39, 30, 48, 27, 46, 29, 49, 31, 52, 37, 59, 33, 56, 40, 64, 35, 60, 38, 65, 42, 68, 43, 71, 44, 73, 45, 75, 51, 82, 47, 79, 112, 50, 84, 53, 88, 54, 90, 57, 94, 55, 93, 61, 100, 58, 98, 62, 103, 63, 105, 67

Offset: 1

Leroy Quet, May 31 2010

This sequence is a permutation of the positive integers.
a(n+1)-a(n) = A175499(n).
Conjecture: the lexicographically earliest permutation of {1,2,...n} for which differences of adjacent numbers are all distinct (cf. A131529) has, for n-->infinity, this sequence as its prefix. - Joerg Arndt, May 27 2012

Joerg Arndt and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1122 terms from Joerg Arndt
Index entries for sequences that are permutations of the natural numbers

Cf. A081145, A175499, A257465 (inverse), A257883, A131388, A131389, A257705.

import Data.List (delete)
a175498 n = a175498_list !! (n-1)
a175498_list = 1 : f 1 [2..] [] where
   f x zs ds = g zs where
     g (y:ys) | diff `elem` ds = g ys
              | otherwise      = y : f y (delete y zs) (diff:ds)
              where diff = y - x
-- Reinhard Zumkeller, Apr 25 2015

a[1] = 1; d[1] = 0; k = 1; z = 10000; zz = 120;
A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
c[k_] := Complement[Range[-z, z], diff[k]];
T[k_] := -a[k] + Complement[Range[z], A[k]];
Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {i, 1, zz}];
u = Table[a[k], {k, 1, zz}]  (* Clark Kimberling, May 13 2015 *)

A175498_list, l, s, b1, b2 = [1,2], 2, 3, set(), set([1])
for n in range(3, 10**5):
    i = s
    while True:
        if not (i in b1 or i-l in b2):
            A175498_list.append(i)
            b1.add(i)
            b2.add(i-l)
            l = i
            while s in b1:
                b1.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 15 2014

More terms from Sean A. Irvine, Jan 27 2011

A257705 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 0.

Original entry on oeis.org

0, 1, 3, 2, 5, 9, 7, 4, 10, 6, 11, 18, 13, 21, 15, 8, 17, 27, 19, 30, 20, 32, 23, 12, 25, 39, 26, 14, 29, 45, 31, 16, 33, 51, 35, 54, 37, 57, 38, 59, 41, 63, 43, 22, 46, 24, 47, 72, 49, 75, 50, 77, 53, 81, 55, 28, 58, 87, 56, 88, 60, 91, 62, 95, 65, 99, 67

Offset: 1

Clark Kimberling, May 12 2015

Rule 1 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
Step 2:  Let h be the least positive integer not in D(k) such that a(k) + h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
Conjecture:  if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0).  Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.
Guide to related sequences:
a(1)  d(1)      (a(n))             (d(n))
0       0      A257705      A131389 except for initial terms
0       1      A257706      A131389 except for initial terms
0       2      A257876      A131389 except for initial terms
0       3      A257877      A257915
1       0      A131388      A131389
1       1      A257878      A131389 except for initial terms
2       0      A257879      A257880
2       1      A257881      A257880 except for initial terms
2       2      A257882      A257918

			a(2) = a(1) + d(2) = 0 + 1 = 1;
a(3) = a(2) + d(3) = 1 + 2 = 3;
a(4) = a(3) + d(4) = 3 + (-1) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A131388, A081145, A257883, A175498.

a[1] = 0; d[1] = 0; k = 1; z = 10000; zz = 120;
A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
c[k_] := Complement[Range[-z, z], diff[k]];
T[k_] := -a[k] + Complement[Range[z], A[k]];
s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
Table[If[Length[s[k]] == 0, {h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {h = Max[s[k]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}], {i, 1, zz}];
u = Table[a[k], {k, 1, zz}] (* A257705 *)
Table[d[k], {k, 1, zz}]     (* A131389 *)

a(k+1) - a(k) = d(k+1) for k >= 1.

Also, a(k) = A131388(n)-1.

A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 8, 5, 11, 7, 12, 19, 14, 22, 16, 9, 18, 28, 20, 31, 21, 33, 24, 13, 26, 40, 27, 15, 30, 46, 32, 17, 34, 52, 36, 55, 38, 58, 39, 60, 42, 64, 44, 23, 47, 25, 48, 73, 50, 76, 51, 78, 54, 82, 56, 29, 59, 88, 57, 89, 61, 92, 63, 96, 66, 100, 68, 35, 70, 106, 72, 37

Offset: 1

Clark Kimberling, Jul 05 2007

nonn

Rule 1 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
Step 2:  Let h be the least positive integer not in D(k) such that a(k) + h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
Conjecture:  if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0).  Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.
See A257705 for a guide to related sequences.

			a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397, A257705, A175498.

(*Program 1 *)
{a, f} = {{1}, {0}}; Do[tmp = {#, # - Last[a]} &[Max[Complement[#, Intersection[a, #]] &[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f,#] && ! MemberQ[a, Last[a] + #]) &]]];
AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {400}];
{a, f} (*{A131388, A131389}; Peter J. C. Moses, May 10 2015*)
(*Program 2 *)
a[1] = 1; d[1] = 0; k = 1; z = 10000; zz = 120;
A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
c[k_] := Complement[Range[-z, z], diff[k]];
T[k_] := -a[k] + Complement[Range[z], A[k]];
s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
Table[If[Length[s[k]] == 0, {h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {h = Max[s[k]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}], {i, 1, zz}];
u = Table[a[k], {k, 1, zz}] (* A131388 *)
Table[d[k], {k, 1, zz}]     (* A131389 *)

a(k+1) - a(k) = d(k+1) for k >= 1.

Revised by Clark Kimberling, May 12 2015

A131393 Conjectured permutation of the positive integers using Rule 2 with a(1)=1.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 8, 5, 11, 7, 12, 19, 14, 22, 16, 9, 18, 28, 20, 31, 21, 33, 24, 13, 26, 40, 27, 42, 30, 15, 32, 48, 34, 17, 35, 54, 38, 58, 39, 60, 37, 59, 41, 64, 44, 23, 47, 25, 50, 76, 52, 79, 53, 81, 56, 29, 61, 90, 62, 92, 63, 94, 57, 91, 55, 88, 49, 84, 51, 87, 46, 83

Offset: 1

Clark Kimberling, Jul 05 2007

Conjecture 1: a( ) is a permutation of the positive integers. Conjecture 2: d( ) is a permutation of the integers. The sequence using Rule 1 ("negative before positive") is A131388.

This sequence was generated using "Rule 2" in a computer program which been lost. The wording of "Rule 2" in the Formula section, although flawed, is retained in case someone can rediscover "Rule 2" and contribute a corrected version. - Clark Kimberling, May 18 2015

			a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4.
The first term that differs from A131388 is a(28)=42.

Table of n, a(n) for n = 1..72

Cf. A131388, A131389, A131390, A131391, A131392, A131394, A131395, A131396, A131397.

The following version of "Rule 2" is defective; see Comments. - Clark Kimberling, May 18 2015

Rule 2 ("positive before negative"): define sequences d( ) and a( ) as follows: d(1)=0, a(1)=1 and for n>=2, d(n) is the least positive integer d such that a(n-1)+d is not among a(1), a(2),...,a(n-1), or, if no such d exists, then d(n) is the greatest negative integer d such that a(n-1)+d is not among a(1), a(2),...,a(n-1). Then a(n)=a(n-1)+d.

A131394 Conjectured permutation of the integers using Rule 2 with A131393(1)=1.

Original entry on oeis.org

0, 1, 2, -1, 3, 4, -2, -3, 6, -4, 5, 7, -5, 8, -6, -7, 9, 10, -8, 11, -10, 12, -9, -11, 13, 14, -13, 15, -12, -15, 17, 16, -14, -17, 18, 19, -16, 20, -19, 21, -23, 22, -18, 23, -20, -21, 24, -22, 25, 26, -24, 27, -26, 28, -25, -27, 32, 29, -28, 30, -29, 31, -37, 34, -36, 33, -39, 35, -33

Offset: 1

Clark Kimberling, Jul 05 2007

Rule 2 is given at A131393. Conjecture: A131394 is a permutation of the integers.

			See A131393.

Table of n, a(n) for n = 1..69

Cf. A131388, A131389, A131390, A131391, A131392, A131393, A131395, A131396, A131397.

This is the sequence d( ) in the formula for A131393.

A131397 Positions of negative integers in A131394, minus 1.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 15, 18, 20, 22, 23, 26, 28, 29, 32, 33, 36, 38, 40, 42, 44, 45, 47, 50, 52, 54, 55, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 79, 81, 83, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 109, 110, 113, 115, 116, 118, 120, 122, 124, 126, 128, 130, 132

Offset: 1

Clark Kimberling, Jul 06 2007

nonn

If A131394 is a permutation of the integers, as conjectured, then A131397 is the complement of A131396.

			The 5th negative integer to occur in A131394 is -5, which occurs at position 13, so that a(5) = 13 - 1 = 12.

Table of n, a(n) for n = 1..65

Cf. A131388, A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396.

a(n) = position of n-th negative integer in A131394, minus 1.

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A131391 Positions of positive integers in A131389, minus 1.

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A131392 Positions of negative integers in A131389, minus 1.

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A175007 Positions of nonnegative terms in A131389.

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A175008 Positions of negative terms in A131389.

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A175498 a(1)=1. a(n) = the smallest positive integer not occurring earlier such that a(n)-a(n-1) doesn't equal a(k)-a(k-1) for any k with 2 <= k <= n-1.

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A257705 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 0.

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A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.

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A131393 Conjectured permutation of the positive integers using Rule 2 with a(1)=1.

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A131394 Conjectured permutation of the integers using Rule 2 with A131393(1)=1.

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