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

0,6

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.

Links

Table of n, a(n) for n=0..90.

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.

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

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

Sequence in context: A271484 A199920 A177995 * A238735 A356006 A258120

Adjacent sequences: A332101 A332102 A332103 * A332105 A332106 A332107

Keyword

nonn,tabl

Author

M. F. Hasler, May 30 2020

Status

approved