A229961 T(n,k) is the number of partitions in each run k of strictly increasing numbers of 2 X 2 squares in the list of partitions of n^2 into squares, where partition sorting order is ascending with larger squares taking higher precedence; irregular triangle T(n,k), 1 <= n, 1 <= k <= A227940(n), read by rows.

1, 2, 3, 1, 5, 2, 1, 7, 5, 2, 3, 1, 1, 10, 7, 5, 3, 1, 6, 3, 1, 2, 3, 1, 1, 13, 11, 8, 6, 4, 2, 9, 7, 4, 2, 5, 3, 1, 7, 4, 2, 3, 4, 2, 1, 17, 14, 12, 10, 8, 5, 3, 1, 13, 10, 8, 6, 4, 1, 9, 6, 4, 2, 5, 2, 1, 10, 8, 6, 4, 1, 6, 4, 2, 2, 4, 2, 8, 5, 3, 1, 4, 1, 1, 4, 2, 1

Offset

1,2

Comments

Row lengths are given by A227940.

Links

Table of n, a(n) for n=1..87.

Christopher Hunt Gribble, C++ program

Example

For n = 4, the 8 partitions of 16 into square parts are:
Partition  Square side
.           1  2  3  4
.
.    1     16  0  0  0
.    2     12  1  0  0
.    3      8  2  0  0
.    4      4  3  0  0
.    5      0  4  0  0
.    6      7  0  1  0
.    7      3  1  1  0
.    8      0  0  0  1
So T(4,1) = 5 as the first runs of 2 X 2 squares is (0,1,2,3,4) from partitions 1 to 5;
T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;
T(4,3) = 1 as the third run is (0) from partition 8.
The irregular triangle begins:
  \  k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  n
  1     1
  2     2
  3     3  1
  4     5  2  1
  5     7  5  2  3  1  1
  6    10  7  5  3  1  6  3  1  2  3  1  1
  7    13 11  8  6  4  2  9  7  4  2  5  3  1  7  4  2  3  4 ...
  8    17 14 12 10  8  5  3  1 13 10  8  6  4  1  9  6  4  2 ...
  9    21 19 16 14 12 10  7  5  3  1 17 15 12 10  8  6  3  1 ...
  10   26 23 21 19 17 14 12 10  8  5  3  1 22 19 17 15 13 10 ...

Crossrefs

Row sums = A037444.

Cf. A227940.

Sequence in context: A066909 A095195 A372640 * A189074 A370484 A255973

Adjacent sequences: A229958 A229959 A229960 * A229962 A229963 A229964

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Oct 04 2013

Status

approved