A257906 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 1.

0, 2, 5, 3, 7, 4, 9, 8, 15, 6, 14, 10, 19, 11, 21, 16, 31, 12, 23, 13, 25, 18, 35, 17, 33, 20, 39, 22, 43, 27, 53, 24, 47, 26, 51, 28, 55, 29, 57, 30, 59, 34, 67, 32, 63, 41, 81, 36, 71, 37, 73, 40, 79, 38, 75, 44, 87, 45, 89, 42, 83, 46, 91, 48, 95, 49, 97

Offset: 1

Author

Clark Kimberling, May 16 2015

Keywords

• nonn
• easy

Comments

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.

Examples

			a(1) = 0, d(1) = 1;
a(2) = 2, d(2) = 2;
a(3) = 5, d(3) = 3;
a(4) = 3, d(4) = -2.
		

Links

• Clark Kimberling (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A257907 (the associated d-sequence, 1 followed by the first differences of this sequence).

Cf. A258105 (inverse to a conjectured permutation of natural numbers which is obtained from this sequence by substituting 1 for the initial value a(1) instead of 0).

Cf. also A257905.

Programs

Haskell

import Data.List ((\\))
a257906 n = a257906_list !! (n-1)
a257906_list = 0 : f [0] [1] where
   f xs@(x:_) ds = g [2 - x .. -1] where
     g [] = y : f (y:xs) (h:ds) where
                  y = x + h
                  (h:_) = [z | z <- [1..] \\ ds, x - z `notElem` xs]
     g (h:hs) | h `notElem` ds && y `notElem` xs = y : f (y:xs) (h:ds)
              | otherwise = g hs
              where y = x + h
-- Reinhard Zumkeller, Jun 03 2015

Mathematica

{a, f} = {{0}, {1}}; 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 *)