A332104 Triangle read by rows in which row n >= 0 lists numbers from 0 to n starting at floor(n/2) and using alternatively larger respectively smaller numbers than the values used so far.

0, 0, 1, 1, 0, 2, 1, 2, 0, 3, 2, 1, 3, 0, 4, 2, 3, 1, 4, 0, 5, 3, 2, 4, 1, 5, 0, 6, 3, 4, 2, 5, 1, 6, 0, 7, 4, 3, 5, 2, 6, 1, 7, 0, 8, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 11, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11, 0, 12

Offset: 0

Author

M. F. Hasler, May 30 2020

Keywords

• nonn
• tabl

Comments

The idea is to cover the range 0..n starting from the center and approaching the limiting values in the most symmetric way, using the smaller value in case of a tie, which leads to each row ending in (the first occurrence of) n.

Contains any sequence of nonnegative integers as a subsequence.

Examples

			The table starts:
Row 0:  0;
Row 1:  0, 1;
Row 2:  1, 0, 2;
Row 3:  1, 2, 0, 3;
Row 4:  2, 1, 3, 0, 4;
Row 5:  2, 3, 1, 4, 0, 5;
Row 5:  3, 2, 4, 1, 5, 0, 6;
Row 6:  3, 4, 2, 5, 1, 6, 0, 7;
Row 7:  4, 3, 5, 2, 6, 1, 7, 0, 8;
Row 8:  4, 5, 3, 6, 2, 7, 1, 8, 0, 9;
Row 9:  5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10;
Row 10: 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 11; ...
Column 1 is floor(n/2) = A004526(n).
The "diagonal" (last element of each row) are the nonnegative integers A001477.
The first subdiagonal is the zero sequence A000004.
The second subdiagonal is the set of positive integers A000027.
The third subdiagonal is "all ones" sequence A000012.
And so on: in alternance, every other subdiagonal is the set of integers >= k, resp., k times the all ones sequence.
		

Crossrefs

Cf. A196199 (concatenate [-n .. n] for n=0, 1, 2...).

Cf. |A128180| = A209279 (based on a very similar idea with positive integers instead).

Programs

Mathematica

Table[Floor[(n + (-1)^(n - k)*k)/2], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 03 2020 *)

PARI

row(n)={ my(m=n\2, M=m, r=[m]); while(#r <= n, r=concat(r, if( n-M > m, M+=1, m-=1))); r}

PARI

T(n,k)=(n+(-1)^(n-k)*k)\2

Formula

If columns are indexed starting from 0:

T(n,0) = floor(n/2) = A004526(n).

T(n,n-2*k) = n-k, for k >= 0.

T(n,n-2*k-1) = k, for k >= 0.

T(n,k) = floor((n+(-1)^(n-k)*k)/2) = (n+k)/2 if n+k even, otherwise floor((n-k)/2).

a(n) = |A128180(n)| - 1.