OFFSET
0,3
COMMENTS
These are row sums of A332080, with terms interpreted as base n+1 numbers.
Rows of A332080 are indeed the nonzero diagonals, read by increasing length, of such triangular matrices. In A332080 the nonzero part of these matrices is seen as a triangle; i.e., rotated by 45 degrees, say. None of the elements of this part may be zero, but they may have (not leading) digits zero. The elements of the vanishing part of the matrix are ignored or considered to have no digit.
a(6) <= 331 and a(7) <= 747, cf. examples in A332080.
EXAMPLE
For n = 0 the matrix and sum of its elements are empty, so a(0) = 0.
For n = 1 the matrix is [1], so a(1) = 1.
For n = 2, a matrix with the required property is given by [ 1, 2; 0, 10 ], where elements are written in base 3. Obviously there can't be a 2 X 2 matrix with this property with smaller sum of elements, so a(2) = 1 + 2 + 3 = 6 (where 3 = 10[3], i.e., 10 in base 3).
For n = 5, one such triangular matrix with minimal sum is given as follows:
1 2 3 4 5
. 15 20 33 44
. . 14 25 30
. . . 10 22
. . . . 11
where all numbers are written in base 6.
One easily checks that no two entries in any row or column have a digit in common.
The sum of these base-6 numbers (e.g., 44[6] = 4*6 + 4 = 28) is a(5) = 159.
There is no such triangle with a smaller sum.
CROSSREFS
KEYWORD
nonn,hard,more,base
AUTHOR
M. F. Hasler, Aug 17 2020
STATUS
approved