A156031 -id:A156031 - cp's OEIS frontend

A167151 a(2n+1) = a(2n) + a(2n-1), a(2n)=least number not yet in the sequence, a(1)=1.

0, 1, 2, 3, 4, 7, 5, 12, 6, 18, 8, 26, 9, 35, 10, 45, 11, 56, 13, 69, 14, 83, 15, 98, 16, 114, 17, 131, 19, 150, 20, 170, 21, 191, 22, 213, 23, 236, 24, 260, 25, 285, 27, 312, 28, 340, 29, 369, 30, 399, 31, 430, 32, 462, 33, 495, 34, 529, 36, 565, 37, 602, 38, 640, 39, 679

Offset: 0

M. F. Hasler, Nov 01 2009

Lexicographically earliest reordering of the nonnegative integers (can be extended by symmetry to a permutation of all integers) such that a(2n+1) = a(2n) + a(2n-1).

Cf. A022941, A143344, A156031.

Cf. A225850 (inverse).

import Data.List (transpose)
a167151 n = a167151_list !! n
a167151_list = 0 : concat (transpose [a005228_list, a030124_list])
-- Reinhard Zumkeller, May 17 2013

a[0] = 0; a[1] = 1;
a[n_?OddQ] := a[n] = a[n - 1] + a[n - 2];
a[n_?EvenQ] := a[n] = For[k = 2, True, k++,
     If[FreeQ[Array[a, n - 1], k], Return[k]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 02 2021 *)

{used=[]; print1(b=0); a=1; for(i=1,99, used=setunion(used,Set(a+=b)); while(setsearch(used,b++), used=setminus(used,Set(b))); print1(", "a", "b))}

a(2n-1) = A005228(n); a(2n) = A030124(n).

A022941 a(n) = a(n-1) + c(n-2) for n >= 3, a( ) increasing, given a(1)=1, a(2)=2; where c( ) is complement of a( ).

Original entry on oeis.org

1, 2, 5, 9, 15, 22, 30, 40, 51, 63, 76, 90, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 354, 383, 414, 446, 479, 513, 548, 584, 621, 659, 698, 739, 781, 824, 868, 913, 959, 1006, 1054, 1103, 1153, 1205, 1258, 1312, 1367, 1423, 1480, 1538

Offset: 1

Clark Kimberling

nonn

Except for a(2), identical to A022940. - Ivan Neretin, Apr 05 2016

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A143344 (first differences), A156031.

Cf. A005228 and references therein.

import Data.List (delete)
a022941 n = a022941_list !! (n-1)
a022941_list = 1 : 2 : f 2 [3..] where
   f x (z:zs) = y : f y (delete y zs) where y = x + z
-- Reinhard Zumkeller, May 17 2013

a[1]:=1: a[2]:=2: c[1]:=3: for n from 2 to 70 do c[n]:=c[n-1]+1: for k from 1 to n do if(c[n]<=a[k])then if(c[n]=a[k])then c[n]:=c[n]+1: fi: break: fi: od: a[n+1]:=a[n]+c[n-1]: od: seq(a[n],n=1..70); # Nathaniel Johnston, May 01 2011

Fold[Append[#1, #1[[-1]] + Complement[Range[Max@#1 + 1], #1][[#2]]] &, {1, 2}, Range[50]] (* Ivan Neretin, Apr 04 2016 *)

a(27) inserted by Nathaniel Johnston, May 01 2011

A143344 First differences of A022941.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77

Offset: 1

N. J. A. Sloane, Nov 01 2009

This is (essentially) the sequence c() mentioned in the definition of A022941.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A022941, A156031.

import Data.List (delete)
a143344 n = a143344_list !! (n-1)
a143344_list = zipWith (-) (tail a022941_list) a022941_list
-- Reinhard Zumkeller, May 17 2013

a[1]:=1: a[2]:=2: c[1]:=3: for n from 2 to 70 do c[n]:=c[n-1]+1: for k from 1 to n do if(c[n]<=a[k])then if(c[n]=a[k])then c[n]:=c[n]+1: fi: break: fi: od: a[n+1]:=a[n]+c[n-1]: od: seq(c[n],n=1..70); # Nathaniel Johnston, May 01 2011

a(26), a(27) corrected by Nathaniel Johnston, May 01 2011

cp's OEIS Frontend

A167151 a(2n+1) = a(2n) + a(2n-1), a(2n)=least number not yet in the sequence, a(1)=1.

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A022941 a(n) = a(n-1) + c(n-2) for n >= 3, a( ) increasing, given a(1)=1, a(2)=2; where c( ) is complement of a( ).

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A143344 First differences of A022941.

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