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%I #62 Jan 06 2025 20:03:56
%S 1,2,2,3,4,3,5,6,6,5,7,8,9,8,7,10,11,12,12,11,10,13,14,15,16,15,14,13,
%T 17,18,19,20,20,19,18,17,21,22,23,24,25,24,23,22,21,26,27,28,29,30,30,
%U 29,28,27,26,31,32,33,34,35,36,35,34,33,32,31,37,38,39,40,41,42,42,41,40,39,38,37
%N Regular symmetric triangle, read by rows, whose left half consists of the positive integers.
%C Left half triangle is A000027 (positive integers) (compare with example triangle):
%C 1;
%C 2;
%C 3, 4;
%C 5, 6;
%C 7, 8, 9;
%C 10, 11, 12;
%C 13, 14, 15, 16;
%C 17, 18, 19, 20;
%C ...
%H Jianrui Xie, <a href="https://arxiv.org/abs/2105.10752">On Symmetric Invertible Binary Pairing Functions</a>, arXiv:2105.10752 [math.CO], 2021. See (6) p. 3 and p. 5
%F T(n,k) = T(n,n-k).
%F T(2*n,n) = (n+1)^2 = A000290(n+1).
%F T(n,0) = T(n,n) = A033638(n+1).
%F From _Yu-Sheng Chang_, May 25 2020: (Start)
%F O.g.f.: F(z,v) = (z/((-z+1)^3*(z+1)) - v^2*z/((-v*z+1)^3*(v*z+1)))/(1-v) + 1/((-z+1)*(-v*z+1)*(-v*z^2+1)).
%F T(n,k) = [v^k] (1/8)*(1-v^(n+1))*(2*(n+1)^2 - 1 - (-1)^n)/(1-v) + (v^(2+n) + (1/2*((sqrt(v)-1)^2*(-1)^n - (sqrt(v)+1)^2))*v^((1/2)*n + 1/2) + 1)/(1-v)^2.
%F T(n,k) = 1 + (1/4)*n*(n+1) + min(k, n-k) + (1/2)*ceiling((1/2)*n). (End)
%F T(n,k) = ((n+k-1)^2 - ((n+k-1) mod 2))/4 + min(n,k) for n and k >= 1, as an array. See Xie. - _Michel Marcus_, May 25 2021
%e Triangle begins:
%e 1;
%e 2, 2;
%e 3, 4, 3;
%e 5, 6, 6, 5;
%e 7, 8, 9, 8, 7;
%e 10, 11, 12, 12, 11, 10;
%e 13, 14, 15, 16, 15, 14, 13;
%e 17, 18, 19, 20, 20, 19, 18, 17;
%e ...
%p A := proc(n,k) ## n >= 0 and k = 0 .. n
%p 1+(1/4)*n*(n+1)+min(k, n-k)+(1/2)*ceil((1/2)*n)
%p end proc: # _Yu-Sheng Chang_, May 25 2020
%t T[n_,k_]:=1+n*(n+1)/4+Min[k,n-k]+Ceiling[n/2]/2;Table[T[n,k],{n,0,11},{k,0,n}]//Flatten (* _James C. McMahon_, Jan 06 2025 *)
%Y Cf. A000027, A000290, A033638 (1st column and right diagonal).
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, May 03 2007
%E Name edited by _Michel Marcus_, May 25 2021