A295508 Triangle read by rows, related to binary partitions of n.

0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 3, 2, 0, 5, 4, 3, 1, 6, 5, 4, 2, 7, 6, 5, 3, 8, 7, 6, 4, 0, 9, 8, 7, 5, 1, 10, 9, 8, 6, 2, 11, 10, 9, 7, 3, 12, 11, 10, 8, 4, 13, 12, 11, 9, 5, 14, 13, 12, 10, 6, 15, 14, 13, 11, 7, 16, 15, 14, 12, 8, 0, 17, 16, 15, 13, 9, 1

Offset: 0

Author

Peter Luschny, Nov 30 2017

Keywords

• nonn
• tabf

Examples

			0;
1,   0;
2,   1,  0;
3,   2,  1;
4,   3,  2,  0;
5,   4,  3,  1;
6,   5,  4,  2;
7,   6,  5,  3;
8,   7,  6,  4, 0;
9,   8,  7,  5, 1;
10,  9,  8,  6, 2;
11, 10,  9,  7, 3;
12, 11, 10,  8, 4;
13, 12, 11,  9, 5;
14, 13, 12, 10, 6;
15, 14, 13, 11, 7;
		

Crossrefs

Cf. A123753.

Programs

Maple

A295508_row := proc(n) local i, s, z; s := n; i := n-1; z := 1;
while 0 <= i do s := s,i; i := i-z; z := z+z od; s end:
seq(A295508_row(n), n=0..17);
# Alternatively after formula:
T := (n, k) -> `if`(k=0, n, n - 2^(k-1)):
L := n -> nops(convert(n, base, 2)) - 0^n:
T_row := n -> seq(T(n,k), k=0..L(n)):
seq(T_row(n), n=0..17);

Formula

Let L(n) = (length of binary representation of n) - 0^n then

T(n, k) = n if k=0 else n - 2^(k-1) for n >= 0 and 0 <= k <= L(n).

Sum_{k=0..L(n)} T(n,k) = A123753(n-1) for n>=1.