A038150 Array of numbers used in exotic ternary numeration system, read by antidiagonals.

1, 2, 3, 4, 6, 8, 5, 11, 16, 21, 7, 14, 29, 42, 55, 9, 19, 37, 76, 110, 144, 10, 24, 50, 97, 199, 288, 377, 12, 27, 63, 131, 254, 521, 754, 987, 13, 32, 71, 165, 343, 665, 1364, 1974, 2584, 15, 35, 84, 186, 432, 898, 1741, 3571, 5168, 6765, 17, 40, 92, 220, 487

Offset

0,2

Links

Table of n, a(n) for n=0..59.

A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.

A. S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284; preprint.

Formula

For n >= 0, A_0^n is the least nonnegative integer not in {A_j^n: 0 <= i < n, j >= 0, A_1^n = 2A_0^n + n, A_j^n = 3A_{j-1}^n - A_{j-2}^n (j >= 2).

a(n,k) = F(2k)*n + F(2k+1)*A026351(n). - Charlie Neder, Feb 07 2019

Example

Top left corner of array:
  1,  3,  8, 21,  55, ...
  2,  6, 16, 42, 110, ...
  4, 11, 29, 76, 199, ...
  5, 14, 37, 97, 254, ...

Mathematica

t[n_, 1] := Floor[(n - 1) GoldenRatio] + 1; t[n_, j_] := Floor[ GoldenRatio^2 t[n, j - 1]] + 1; Table[ t[n - m + 1, m], {n, 11}, {m, n}] // Flatten (* Birkas Gyorgy, Apr 15 2011; modified by Robert G. Wilson v, Apr 15 2011 *)

Crossrefs

Rows give A001906, A025169, A002878.

Columns give A026351, A047924, A047925.

Main diagonal gives A047923.

Cf. A026351, A035506.

Sequence in context: A257802 A369271 A225850 * A182831 A257725 A257690

Adjacent sequences: A038147 A038148 A038149 * A038151 A038152 A038153

Keyword

nonn,tabl,nice,easy

Author

Aviezri S. Fraenkel

Extensions

More terms from Naohiro Nomoto, Jun 07 2001

Status

approved