The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). LINKS Clark Kimberling and Peter J. C. Moses, Complementary Equations with Advanced Subscripts, J. Int. Seq. 24 (2021) Article 21.3.3. EXAMPLE 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 MATHEMATICA 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 *) CROSSREFS 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). Sequence in context: A026183 A026199 A026207 * A261645 A193724 A266407 Adjacent sequences:  A309154 A309155 A309156 * A309158 A309159 A309160 KEYWORD nonn,tabf,easy AUTHOR Clark Kimberling, Jul 15 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 16 12:45 EDT 2021. Contains 345057 sequences. (Running on oeis4.)