A243070 Square array read by antidiagonals: rows are successively recursivized versions of Bulgarian solitaire operation (starting from the usual "first order" version, A242424), as applied to the partitions listed in A112798.

1, 2, 1, 4, 2, 1, 3, 4, 2, 1, 6, 3, 4, 2, 1, 6, 8, 3, 4, 2, 1, 10, 6, 8, 3, 4, 2, 1, 5, 12, 6, 8, 3, 4, 2, 1, 12, 5, 16, 6, 8, 3, 4, 2, 1, 9, 9, 5, 16, 6, 8, 3, 4, 2, 1, 14, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 10, 20, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 22, 10, 24, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 15, 28, 10, 32, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 18, 18, 40, 10, 32, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1

Offset

1,2

Comments

The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

Please see comments and references in A242424 for more information about Bulgarian Solitaire.

Each row is a A241909-conjugate of the corresponding row in A243060.

Rows in both arrays converge towards A122111.

All the terms in column n are multiples of A105560(n).

The rows of this table (i.e., the corresponding functions) preserve A056239.

First point where row k differs from row k of A243060 seems to be A000040(k+2): primes from five onward: 5, 7, 11, 13, 17, 19, 23, 29, 31, ... and these seem to be also the points where that row differs for the first time from A122111.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the square array

Formula

A(1,col) = A242424(col), otherwise, when row > 1, A(row,col) = A000040(A001222(col)) * A(row-1, A064989(col)).

Example

The top left corner of the array is:
  1,  2,  4,  3,  6,  6, 10,  5, 12,  9, 14, 10, 22, 15, 18, ...
  1,  2,  4,  3,  8,  6, 12,  5,  9, 12, 20, 10, 28, 18, 18, ...
  1,  2,  4,  3,  8,  6, 16,  5,  9, 12, 24, 10, 40, 24, 18, ...
  1,  2,  4,  3,  8,  6, 16,  5,  9, 12, 32, 10, 48, 24, 18, ...
  1,  2,  4,  3,  8,  6, 16,  5,  9, 12, 32, 10, 64, 24, 18, ...

Prog

(Scheme)
(define (A243070 n) (A243070bi (A002260 n) (A004736 n)))
(define (A243070bi row col) (cond ((<= col 1) col) ((= 1 row) (A242424 col)) (else (* (A000040 (A001222 col)) (A243070bi (- row 1) (A064989 col))))))

Crossrefs

Row 1: A242424, Row 2: A243072, Row 3: A243073.

Rows converge towards A122111.

Cf. A243060, A241909, A112798, A105560, A000040, A001222, A064989.

Sequence in context: A378198 A205395 A367647 * A243060 A286321 A349192

Adjacent sequences: A243067 A243068 A243069 * A243071 A243072 A243073

Keyword

nonn,tabl

Author

Antti Karttunen, May 29 2014

Status

approved