A257884 -id:A257884 - cp's OEIS frontend

A257883 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 0.

0, 1, 3, 2, 5, 9, 4, 10, 6, 11, 8, 15, 7, 16, 14, 22, 12, 23, 17, 27, 13, 25, 18, 31, 19, 33, 20, 35, 24, 40, 21, 38, 29, 47, 26, 45, 28, 48, 30, 51, 36, 58, 32, 55, 39, 63, 34, 59, 37, 64, 41, 67, 42, 70, 43, 72, 44, 74, 50, 81, 46, 78, 111, 49, 83, 52, 87

Offset: 1

Clark Kimberling, May 13 2015

Algorithm: 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). Let h be the least integer > -a(k) such that h is not in D(k) and 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 repeat inductively.
Conjecture: if a(1) is a 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      A257883      A175499 except for initial terms
0      1      A257884      A175499
0      2      A257885      A257902
0      3      A257903      A257904
1      0      A175498      A175499 except for first term
1      1      A257905      A175499
2      0      A257908      A257909
2      1      A257910      A257909 except for initial terms
2      2      A257911      A257912

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

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

Cf. A081145, A175498, A257705.

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]]
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}]  (* A257883, = -1 + A175498 *)
Table[d[k], {k, 1, zz}] (* A175499 except that here first term is 0 *)

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

Also, A257883(n) = -1 + A175498(n) for n >= 1.

A175499 a(n) = A175498(n+1)-A175498(n).

Original entry on oeis.org

1, 2, -1, 3, 4, -5, 6, -4, 5, -3, 7, -8, 9, -2, 8, -10, 11, -6, 10, -14, 12, -7, 13, -12, 14, -13, 15, -11, 16, -19, 17, -9, 18, -21, 19, -17, 20, -18, 21, -15, 22, -26, 23, -16, 24, -29, 25, -22, 27, -23, 26, -25, 28, -27, 29, -28, 30, -24, 31, -35, 32, 33, -62, 34, -31, 35, -34, 36, -33, 37, -39, 38, -32, 39, -42, 40, -36

Offset: 1

Leroy Quet, May 31 2010

sign

No integer occurs in this sequence more than once, by definition. Is this sequence a permutation of the nonzero integers?

Chai Wah Wu, Table of n, a(n) for n = 1..4999

Cf. A175498, A257884, A131389.

a175499 n = a175499_list !! (n-1)
a175499_list = zipWith (-) (tail a175498_list) a175498_list
-- Reinhard Zumkeller, Apr 25 2015

a[1] = 0; d[1] = 1; 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}]  (* A257884 *)
Table[d[k], {k, 1, zz}] (* A175499 *)
(* Clark Kimberling, May 13 2015 *)

A175499_list, l, s, b = [1], 2, 3, set()
for n in range(2, 10**2):
    i, j = s, s-l
    while True:
        if not (i in b or j in A175499_list):
            A175499_list.append(j)
            b.add(i)
            l = i
            while s in b:
                b.remove(s)
                s += 1
            break
        i += 1
        j += 1 # Chai Wah Wu, Dec 15 2014

More terms from Sean A. Irvine, Jan 27 2011

cp's OEIS Frontend

A257883 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 0.

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