A067576 - OEIS

A067576

Array T(i,j) read by downward antidiagonals, where T(i,j) is the j-th term whose binary expansion has i 1's.

1, 2, 3, 4, 5, 7, 8, 6, 11, 15, 16, 9, 13, 23, 31, 32, 10, 14, 27, 47, 63, 64, 12, 19, 29, 55, 95, 127, 128, 17, 21, 30, 59, 111, 191, 255, 256, 18, 22, 39, 61, 119, 223, 383, 511, 512, 20, 25, 43, 62, 123, 239, 447, 767, 1023, 1024, 24, 26, 45, 79, 125, 247, 479, 895, 1535, 2047

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OFFSET

1,2

COMMENTS

This is a permutation of the positive integers; the inverse permutation is A356419. - Jianing Song, Aug 06 2022

Row i shows the numbers m, in increasing order, such that 2^i divides binomial(2*m, m) and 2^(i+1) does not. Let T'(i, j) = 1 + T(i, j) for all i and j. Conjecture: every column of T' is eventually geometric with common ratio 2. Assuming the conjecture is true, let d(k) be the number in column k of the row in which this doubling begins; then d(1) = 1, d(2) = 1, d(3) = 2, d(4) = 3, d(5) = 4. - Clark Kimberling, Nov 08 2025

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

Array begins:

j=1 j=2 j=3 j=4 j=5 j=6

i=1: 1, 2, 4, 8, 16, 32, ...

i=2: 3, 5, 6, 9, 10, 12, ...

i=3: 7, 11, 13, 14, 19, 21, ...

i=4: 15, 23, 27, 29, 30, 39, ...

i=5: 31, 47, 55, 59, 61, 62, ...

i=6: 63, 95, 111, 119, 123, 125, ...