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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. 2

%I

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

%C Row sums = odd integers repeated: (1, 1, 3, 3, 5, 5,...).

%C Eigensequence of the triangle = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401,...).

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

%F From _Boris Putievskiy_, Jan 09 2013: (Start)

%F a(n) = 2*A101688(n)-A023531(n).

%F a(n) = 2*floor((2*A002260(n)+1)/(A003056(n)+3))*A002260(n)-A023531(n).

%F 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 First few rows of the triangle =

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 0, 2, 1;

%e 0, 0, 2, 2, 1;

%e 0, 0, 0, 2, 2, 1;

%e 0, 0, 0, 2, 2, 2, 1;

%e 0, 0, 0, 0, 2, 2, 2, 1;

%e 0, 0, 0, 0, 2, 2, 2, 2, 1;

%e 0, 0, 0, 0, 0, 2, 2, 2, 2, 1;

%e 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;

%e 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;

%e ...

%t rows = 11;

%t A = Array[Which[#1 == 1, 1, #1 <= #2, 2, True, 0]&, {rows, rows}];

%t Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* _Jean-Fran├žois Alcover_, Aug 08 2018 *)

%Y Cf. A101688, A168314, A168315

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, Nov 22 2009

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Last modified September 18 04:14 EDT 2020. Contains 337165 sequences. (Running on oeis4.)