A257984 -id:A257984 - cp's OEIS frontend

A257987 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 2 and d(1) = 2.

2, 3, 6, 4, 8, 5, 10, 9, 17, 7, 13, 25, 11, 21, 12, 23, 15, 29, 14, 27, 16, 31, 18, 35, 19, 37, 20, 39, 32, 26, 22, 43, 24, 47, 42, 30, 59, 28, 55, 33, 65, 36, 71, 34, 67, 40, 79, 38, 75, 41, 81, 45, 89, 44, 87, 48, 95, 46, 91, 49, 97, 50, 99, 51, 101, 57

Offset: 1

Clark Kimberling, Jun 02 2015

Rule 3 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 least 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.
See A257905 for a guide to related sequences and conjectures.

			a(1) = 2, d(1) = 2;
a(2) = 3, d(2) = 1;
a(3) = 6, d(3) = 3;
a(4) = 4, d(4) = -2.

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

Cf. A257905, A257909.

{a, f} = {{2}, {2}}; Do[tmp = {#, # - Last[a]} &[Min[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]]], {120}]; {a, f} (* Peter J. C. Moses, May 14 2015 *)

a(n) - a(n-1) = A257984(n) for n >= 2.

A258048 Nonhomogeneous Beatty sequence: a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97

Offset: 0

Clark Kimberling, Jun 15 2015

See A257984.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Aviezri S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27.

Cf. A257984 (complement), A246046, A062380, A258833.

Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)
Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

cp's OEIS Frontend

A257987 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 2 and d(1) = 2.

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