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

0,4

Links

Table of n, a(n) for n=0..76.

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.

Example

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;

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);

Crossrefs

Cf. A123753.

Sequence in context: A019586 A257962 A176095 * A260672 A063942 A263405

Adjacent sequences: A295505 A295506 A295507 * A295509 A295510 A295511

Keyword

nonn,tabf

Author

Peter Luschny, Nov 30 2017

Status

approved