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 A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0. 17
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Clark Kimberling, Table of n, a(n) for n = 1..1000 FORMULA a(k+1) - a(k) = d(k+1) for k >= 1. EXAMPLE a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4. MATHEMATICA (*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[]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] + #]) &]]]; AppendTo[a, tmp[]]; AppendTo[f, tmp[]], {400}]; {a, f} (*{A131388, A131389}; Peter J. C. Moses, May 10 2015*) (*Program 2 *) a = 1; d = 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 *) CROSSREFS Cf. A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397, A257705, A175498. Sequence in context: A318452 A083673 A327120 * A131393 A216476 A002326 Adjacent sequences:  A131385 A131386 A131387 * A131389 A131390 A131391 KEYWORD nonn AUTHOR Clark Kimberling, Jul 05 2007 EXTENSIONS Revised by Clark Kimberling, May 12 2015 STATUS approved

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Last modified October 22 13:19 EDT 2019. Contains 328318 sequences. (Running on oeis4.)