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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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, July 12 2019 *)

CROSSREFS

Cf. A304799 (row 0).

Sequence in context: A021369 A242304 A227415 * A298042 A051430 A185510

Adjacent sequences:  A309153 A309154 A309155 * A309157 A309158 A309159

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jul 15 2019

STATUS

approved

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Last modified January 20 07:59 EST 2020. Contains 331081 sequences. (Running on oeis4.)