A309156 Rectangular array by antidiagonals: row c is the solution sequence (a(n)) of the complementary equation a(n) = b(n) + b(2n) + c, for c >= 0. See Comments.

2, 7, 3, 10, 7, 4, 14, 11, 7, 5, 18, 15, 11, 8, 6, 23, 19, 16, 12, 9, 7, 26, 23, 20, 16, 12, 10, 8, 31, 27, 24, 20, 16, 13, 11, 9, 34, 31, 28, 24, 20, 17, 14, 12, 10, 38, 35, 31, 29, 25, 21, 17, 15, 13, 11, 43, 39, 36, 32, 29, 25, 21, 18, 16, 14, 12, 46, 43

Offset

0,1

Comments

See A304799 for the generation of the top row (indexed as row 0) as the sequence (a(n)), which, along with (b(n)) = A304800, is the unique solution of a(n) = b(n) + b(2n) + 0.

Regarding row 1, it is easy to prove that the solution (a(n)) of a(n) = b(n) + b(2n) + 1 is given by a(n) = 4n+3. Conjecture: this is the only linearly recurrent row.

Using the notation w(m, n) for n-th term in row m, for m >= 0 and n >= 0, note that a(m+1,n) - a(m,n) is not always in {-1, 0, 1}; e.g. a(6, 60) = 243 and a(5, 60) = 241.

Links

Table of n, a(n) for n=0..67.

Example

Northwest corner:
2    7    10   14   18   23   26   31   34   38
3    7    11   15   19   23   27   31   35   39
4    7    11   16   20   24   28   31   36   40
5    8    12   16   20   24   29   32   36   41
6    9    12   16   20   25   29   33   37   41
7    10   13   17   21   25   29   33   38   41
8    11   14   17   21   25   29   34   38   42

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
h = 1; k = 2;
Table[a[c] = {}; b = {1};
  AppendTo[a[c], c + mex[Flatten[{a[c], b}], 1]];
  Do[Do[AppendTo[b, mex[Flatten[{a[c], b}], Last[b]]], {k}];
  AppendTo[a[c],
  c + Last[b] + b[[1 + (Length[b] - 1)/k h]]]; , {100}], {c, 0, 20}];
w[n_, k_] := a[n - 1][[k]];
Table[w[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten
(* Peter J. C. Moses, Jul 12 2019 *)

Crossrefs

Cf. A304799 (row 0).

Sequence in context: A242304 A227415 A361207 * A298042 A051430 A185510

Adjacent sequences: A309153 A309154 A309155 * A309157 A309158 A309159

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jul 15 2019

Status

approved