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A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

%I A224380 #16 Sep 08 2022 08:46:04
%S A224380 11,19,27,35,51,59,67,99,115,123,131,195,227,243,251,259,387,451,483,
%T A224380 499,507,515,771,899,963,995,1011,1019,1027,1539,1795,1923,1987,2019,
%U A224380 2035,2043,2051,3075,3587,3843,3971,4035,4067,4083,4091,4099,6147,7171,7683,7939,8067,8131,8163,8179
%N A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.
%C A224380 The table has row labels 2^n - 1 and column labels 2^(m+2). The table entry is row*col + 3. 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 A224380 A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
%C A224380 +3  |    8    16    32    64   128    256    512 ...
%C A224380 ----|-------------------------------------------
%C A224380 1   |   11    19    35    67   131    259    515
%C A224380 3   |   27    51    99   195   387    771   1539
%C A224380 7   |   59   115   227   451   899   1795   3587
%C A224380 15  |  123   243   483   963  1923   3843   7683
%C A224380 31  |  251   499   995  1987  3971   7939  15875
%C A224380 63  |  507  1011  2019  4035  8067  16131  32259
%C A224380 127 | 1019  2035  4067  8131 16259  32515  65027
%C A224380 ...
%C A224380 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 an alternating series of 1's and 3's with the last difference in the pattern m. The number of alternating 1's and 3's in the pattern is 2^(j+1) - 1, where j is the column index.
%C A224380 As an example consider A(1) which is 11, the sequence B(n) where i XOR 10 = i - 10 starts as 10, 11, 14, 15, 26, 27, 30, 31, 42, ... (A214864) with successive differences of 1, 3, 1, 11.
%C A224380 Main diagonal is A191341, the largest k such that k-1 and k+1 in binary representation have the same number of 1's and 0's
%H A224380 Brad Clardy, <a href="/A224380/b224380.txt">Table of n, a(n) for n = 1..1000</a>
%F A224380 a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 2) + 3 for n>=1.
%o A224380 (Magma)
%o A224380 //program generates values in a table form,row labels of 2^i -1
%o A224380 for i:=1 to 10 do
%o A224380     m:=2^i - 1;
%o A224380     m, [ m*2^n +1 : n in [1..10]];
%o A224380 end for;
%o A224380 //program generates sequence in lexicographic ordering of A057555, read
%o A224380 //along antidiagonals from top. Primes in the sequence are marked with *.
%o A224380 for i:=2 to 18 do
%o A224380     for j:=1 to i-1 do
%o A224380        m:=2^j -1;
%o A224380        k:=m*2^(2+i-j) + 3;
%o A224380        if IsPrime(k) then k, "*";
%o A224380           else k;
%o A224380        end if;;
%o A224380     end for;
%o A224380 end for;
%Y A224380 Cf. A057555(lexicographic ordering), A214864(example), A224195.
%Y A224380 Rows: A062729(i=1), A147595(2 n>=5), A164285(3 n>=3).
%Y A224380 Cols: A168616(j=1 n>=4).
%Y A224380 Diagonal: A191341.
%K A224380 tabl,nonn
%O A224380 1,1
%A A224380 _Brad Clardy_, Apr 05 2013