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A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

%I A224701 #14 Sep 08 2022 08:46:04
%S A224701 21,37,53,69,101,117,133,197,229,245,261,389,453,485,501,517,773,901,
%T A224701 965,997,1013,1029,1541,1797,1925,1989,2021,2037,2053,3077,3589,3845,
%U A224701 3973,4037,4069,4085,4101,6149,7173,7685,7941,8069,8133,8165,8181,8197,12293,14341,15365,15877,16133
%N A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.
%C A224701 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 antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:
%C A224701 A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
%C A224701 +5  |   16    32    64   128   256    512   1024 ...
%C A224701 ----|-------------------------------------------
%C A224701 1   |   21    37    69   133   261    517   1029
%C A224701 3   |   53   101   197   389   773   1541   3077
%C A224701 7   |  117   229   453   901  1797   3589   7173
%C A224701 15  |  245   485   965  1925  3845   7685  15365
%C A224701 31  |  501   997  1989  3973  7941  15877  31749
%C A224701 63  | 1013  2021  4037  8069 16133  32261  64517
%C A224701 127 | 2037  4069  8133 16261 32517  65029 130053
%C A224701 ...
%C A224701 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
%C A224701   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.
%C A224701 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.
%C A224701 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.
%H A224701 Brad Clardy, <a href="/A224701/b224701.txt">Table of n, a(n) for n = 1..1000</a>
%F A224701 a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 3) + 5 for n>=1.
%o A224701 (Magma)
%o A224701 //program generates values in a table form, row labels of 2^i -1
%o A224701 for i:=1 to 10 do
%o A224701     m:=2^i - 1;
%o A224701     m, [ m*2^(n+3) +5 : n in [1..10]];
%o A224701 end for;
%o A224701 //program generates sequence in lexicographic ordering of A057555, read
%o A224701 //along antidiagonals from top. Primes in the sequence are marked with *.
%o A224701 for i:=2 to 18 do
%o A224701     for j:=1 to i-1 do
%o A224701        m:=2^j -1;
%o A224701        k:=m*2^(3+i-j) + 5;
%o A224701        if IsPrime(k) then k, "*";
%o A224701           else k;
%o A224701        end if;
%o A224701     end for;
%o A224701 end for;
%Y A224701 Cf. A057555 (lexicographic ordering).
%Y A224701 Rows: A168614(i=1), n>=4.
%Y A224701 Cols: A220087(j=2), n>=6.
%K A224701 nonn,tabl
%O A224701 1,1
%A A224701 _Brad Clardy_, Apr 16 2013