(MAGMAMagma)

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Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

//along anti-diagonals antidiagonals from top. Primes in the sequence are marked with *.

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Table read by anti-diagonals antidiagonals of numbers of form (2^n -1)*2^(m+3) + 5 where n>=1, m>=1.

The table has row labels 2^n - 1 and column labels 2^(m+3). The table entry is row*col + 5. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the anti-diagonals antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:

All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is a repeating series

let of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m be a member . The total number of A1's and 5's in the pattern is 2^(nj+2), - 1, where j is the column index.

if a As an example, consider A(1), which is 21; the sequence B(n) = all i such that where i XOR (m - 1) 20 = i - (m - 20 starts as 20, 21, 22, 23, 28, 29, 30, 31, 52, ... with successive differences of 1, 1, 1, 5, 1, 1, 1), then, 21.

for A(2), which is 37, the sequence B(n) where i XOR 36 = i - 36 starts as 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 100, ... with successive differences between successive members of B(n) is a repeating series1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 37.

of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m.

The total number of 1's and 5's in the pattern is 2^(j+2) - 1, where

j is the column index.

As an example consider A(1) which is 21,

the sequence B(n) where i XOR 20 = i - 20 starts as:

20, 21, 22, 23, 28, 29, 30, 31, 52, ...

with successive differences of:

1, 1, 1, 5, 1, 1, 1, 21.

for A(2) which is 37,

the sequence B(n) where i XOR 36 = i - 36 starts as:

36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 100,...

with successive differences of:

1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 37

Cf. A057555 (lexicographic ordering).

Rows : A168614(i=1) , n>=4.

Cols. : A220087(j=2) , n>=6.

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Brad Clardy, <a href="/A224701/b224701.txt">Table of n, a(n) for n = 1..9991000</a>

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The table has row labels 2^n - 1 and column labels 2^(m+3). The table entry is row*col + 5. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the anti-diagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:

A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...

let m be a member of A(n),

if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then

the differences between successive members of B(n) is a repeating series

of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m.

The total number of 1's and 5's in the pattern is 2^(j+2) - 1, where

j is the column index.

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 3) + 5 for n>=1.

Cf. A057555(lexicographic ordering)