

A257906


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


4



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
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OFFSET

1,2


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.


LINKS

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


EXAMPLE

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


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 *)


PROG

(Scheme, with memoizationmacro definec)
(definec (A257906 n) (if (= 1 n) 0 (+ (A257906 ( n 1)) (A257907 n))))
;; Needs also code from A257907.  Antti Karttunen, May 20 2015
(Haskell)
import Data.List ((\\))
a257906 n = a257906_list !! (n1)
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


CROSSREFS

Cf. A257907 (the associated dsequence, 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.
Sequence in context: A253924 A141410 A333811 * A257983 A210770 A227688
Adjacent sequences: A257903 A257904 A257905 * A257907 A257908 A257909


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 16 2015


STATUS

approved



