A067017 -id:A067017 - cp's OEIS frontend

A067016 Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = max_i (a(i)+a(n-i)).

1, 4, 3, 2, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124, 127, 128

Offset: 0

N. J. A. Sloane, Feb 17 2002

R. K. Guy, Unsolved Problems in Number Theory, E27.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A067017, A067018.

a067016 n = a067016_list !! n
a067016_list = [1,4,3,2] ++ f [2,3,4,1] where
   f xs = maxi : f (maxi : xs) where
     maxi = maximum $ zipWith (+) xs (reverse xs)
-- Reinhard Zumkeller, May 05 2012

First differences are ultimately periodic.

Conjecture: a(n) = (-3+(-1)^n+4*n)/2 for n>3. G.f.: -(2*x^6 -2*x^5 -6*x^4 +4*x^3 +2*x^2 -3*x -1) / ((x -1)^2*(x +1)). - Colin Barker, Apr 01 2013

More terms from John W. Layman, Feb 20 2002

A067018 Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = mex_i (nim-sum a(i)+a(n-i)), where mex means smallest nonnegative missing number.

Original entry on oeis.org

1, 4, 3, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0

Offset: 0

N. J. A. Sloane, Feb 17 2002

Nim-sum is addition in base 2 without carry (XOR the binary expansions).

			a(5) = mex{1 xor 0, 4 xor 2, 3 xor 3, etc. (duplicates)} = mex{1 xor 0, 100 xor 10, 11 xor 11} (in base 2) = mex{1, 6, 0} = 2

R. K. Guy, Unsolved Problems in Number Theory, E27.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A067016, A067017.

import Data.Bits (xor)
import Data.List ((\\))
a067018 n = a067018_list !! n
a067018_list =  [1,4,3,2] ++ f [2,3,4,1] where
  f xs = mexi : f (mexi : xs) where
    mexi = head $ [0..] \\ zipWith xor xs (reverse xs) :: Integer
-- Reinhard Zumkeller, May 05 2012

More terms from John W. Layman, Feb 20 2002

A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 5, 3, 4, 12, 6, 7, 20, 8, 9, 26, 10, 11, 33, 13, 14, 41, 15, 16, 47, 17, 18, 54, 19, 21, 61, 22, 23, 68, 24, 25, 75, 27, 28, 83, 29, 30, 89, 31, 32, 96, 34, 35, 104, 36, 37, 110, 38, 39, 117, 40, 42, 124, 43, 44, 131, 45, 46, 138, 48, 49, 146, 50, 51

Offset: 1

Clark Kimberling, Jul 15 2019

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for these conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).

			c(1) = a(1) + b(2) > = 1 + 3, so that
a(2) = mex{1,2} = 3;
b(2) = mex{1,2,3} = 4;
c(1) = 5.
Then c(2) = a(2) + b(4) >= 3 + 8, so that
a(3) = 6, b(3) = 7;
a(4) = 8, b(4) = 9;
c(2) = a(2) + b(4) = 3 + 9 = 12.
   n    a(n) b(n) c(n)
  --------------------
   1      1    2    5
   2      3    4   12
   3      6    7   20
   4      8    9   26
   5     10   11   33
   6     13   14   41
   7     15   16   47
   8     17   18   54
   9     19   21   61
  10     22   23   68

Clark Kimberling and Peter J. C. Moses, Complementary Equations with Advanced Subscripts, J. Int. Seq. 24 (2021) Article 21.3.3.

Cf. A326663 (3rd column),
A101544 solves c(n) = a(n) + b(n),
A326661 solves c(n) = a(n) + b(3n),
A326662 solves c(n) = a(2n) + b(2n).

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 1; k = 2;
Do[Do[AppendTo[a,
  mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
{a, b, c} // ColumnForm
a = Take[a, Length[c]]; b = Take[b, Length[c]];
Flatten[Transpose[{a, b, c}]] (* Peter J. C. Moses, Jul 04 2019 *)

A326661 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(3n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 7, 3, 4, 16, 5, 6, 25, 8, 9, 35, 10, 11, 43, 12, 13, 52, 14, 15, 61, 17, 18, 71, 19, 20, 79, 21, 22, 88, 23, 24, 97, 26, 27, 107, 28, 29, 115, 30, 31, 124, 32, 33, 133, 34, 36, 142, 37, 38, 151, 39, 40, 160, 41, 42, 169, 44, 45, 179, 46, 47, 187, 48

Offset: 1

Clark Kimberling, Jul 16 2019

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for the following conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).

			c(1) = a(1) + b(3) >= 1 + 6, so that b(1) = mex{1} = 2; a(2) = mex{1,2} = 3; b(2) = mex{1,2,3} = 4; a(3)= mex{1,2,3,4} = 5, a(4) = mex{1,2,3,4,5} = 6, c(1) = 7.
n           a(n)      b(n)     c(n)
---------------------------
1             1        2        7
2             3        4       16
3             5        6       25
4             8        9       35
5            10       11       43
6            12       13       52
7            14       15       61
8            17       18       74
9            19       20       79
10           21       22       88

Cf. A309157, A326662, A326664.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 1; k = 3;
Do[Do[AppendTo[a,
  mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
{a, b, c} // ColumnForm
a = Take[a, Length[c]]; b = Take[b, Length[c]];
Flatten[Transpose[{a, b, c}]](* Peter J. C. Moses, Jul 04 2019 *)

Replaced a(0)->a(1) in NAME. - R. J. Mathar, Jun 19 2021

A326662 Rectangular array in 3 columns that solve the complementary equation c(n) = a(2n) + b(2n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 7, 3, 4, 17, 5, 6, 25, 8, 9, 34, 10, 11, 43, 12, 13, 53, 14, 15, 61, 16, 18, 71, 19, 20, 79, 21, 22, 89, 23, 24, 97, 26, 27, 106, 28, 29, 115, 30, 31, 125, 32, 33, 133, 35, 36, 142, 37, 38, 151, 39, 40, 161, 41, 42, 169, 44, 45, 178, 46, 47, 187, 48

Offset: 1

Clark Kimberling, Jul 16 2019

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for the following conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).

			c(1) = a(2) + b(2) >= 3 + 4, so that b(1) = mex{1} = 2; a(2) = mex{1,2} = 3; b(2) = mex{1,2,3} = 4; a(3)= mex{1,2,3,4} = 5, a(4) = mex{1,2,3,4,5} = 6, c(1) = 7.
n           a(n)      b(n)     c(n)
-----------------------------------
1             1        2        7
2             3        4       17
3             5        6       25
4             8        9       34
5            10       11       43
6            12       13       53
7            14       15       61
8            16       18       71
9            19       20       79
10           21       22       89

Cf. A309157, A326661.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 2; k = 2;
Do[Do[AppendTo[a,
   mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
{a, b, c} // ColumnForm
a = Take[a, Length[c]]; b = Take[b, Length[c]];
Flatten[Transpose[{a, b, c}]](* Peter J. C. Moses, Jul 04 2019 *)

Replaced a(0)->a(1) in NAME. - R. J. Mathar, Jun 19 2021

cp's OEIS Frontend

A067016 Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = max_i (a(i)+a(n-i)).

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A067018 Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = mex_i (nim-sum a(i)+a(n-i)), where mex means smallest nonnegative missing number.

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A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

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A326661 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(3n), where a(1) = 1; see Comments.

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A326662 Rectangular array in 3 columns that solve the complementary equation c(n) = a(2n) + b(2n), where a(1) = 1; see Comments.

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Comments

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Programs

Mathematica

Extensions