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A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.

%I A168313 #22 Aug 08 2018 04:34:08
%S A168313 1,0,1,0,2,1,0,0,2,1,0,0,2,2,1,0,0,0,2,2,1,0,0,0,2,2,2,1,0,0,0,0,2,2,
%T A168313 2,1,0,0,0,0,2,2,2,2,1,0,0,0,0,0,2,2,2,2,1,0,0,0,0,0,2,2,2,2,2,1
%N A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
%C A168313 Row sums = odd integers repeated: (1, 1, 3, 3, 5, 5,...).
%C A168313 Eigensequence of the triangle = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401,...).
%H A168313 Boris Putievskiy, <a href="https://arxiv.org/abs/1212.2732">Transformations (of) Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
%F A168313 From _Boris Putievskiy_, Jan 09 2013: (Start)
%F A168313 a(n) = 2*A101688(n)-A023531(n).
%F A168313 a(n) = 2*floor((2*A002260(n)+1)/(A003056(n)+3))*A002260(n)-A023531(n).
%F A168313 a(n) = 2*floor((2*n-t*(t+1)+1)/(t+3))*(n-t*(t+1)/2) - floor((sqrt(8*n+1)-1)/2) + t, where t = floor((-1+sqrt(8*n-7))/2). (End)
%e A168313 First few rows of the triangle =
%e A168313 1;
%e A168313 0, 1;
%e A168313 0, 2, 1;
%e A168313 0, 0, 2, 1;
%e A168313 0, 0, 2, 2, 1;
%e A168313 0, 0, 0, 2, 2, 1;
%e A168313 0, 0, 0, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 2, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 0, 2, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
%e A168313 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
%e A168313 ...
%t A168313 rows = 11;
%t A168313 A = Array[Which[#1 == 1, 1, #1 <= #2, 2, True, 0]&, {rows, rows}];
%t A168313 Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Aug 08 2018 *)
%Y A168313 Cf. A101688, A168314, A168315
%K A168313 nonn,tabl
%O A168313 1,5
%A A168313 _Gary W. Adamson_, Nov 22 2009