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%I #30 Jan 08 2022 23:51:30
%S 1,3,2,5,4,3,7,6,5,4,9,8,7,6,5,11,10,9,8,7,6,13,12,11,10,9,8,7,15,14,
%T 13,12,11,10,9,8,17,16,15,14,13,12,11,10,9,19,18,17,16,15,14,13,12,11,
%U 10,21,20,19,18,17,16,15,14,13,12,11
%N Triangle T(n,k) = 2*n-k, read by rows.
%C From _Boris Putievskiy_, Jan 24 2013: (Start)
%C Table T(n,k) = n+2*k-2 n, k > 0, read by antidiagonals.
%C General case A209304. Let m be natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. Every next column is formed from previous shifted by m elements.
%C For m=0 the result is A002260,
%C for m=1 the result is A002024,
%C for m=2 the result is A128076,
%C for m=3 the result is A131914,
%C for m=4 the result is A209304. (End)
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [Of] Integer Sequences And Pairing Functions</a>, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
%F Matrix product A128064 * A004736 as infinite lower triangular matrices.
%F From _Boris Putievskiy_, Jan 24 2013: (Start)
%F For the general case:
%F a(n) = m*A003056 -(m-1)*A002260.
%F a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n), where t=floor((-1+sqrt(8*n-7))/2).
%F For m = 2:
%F a(n) = 2*A003056 -A002260.
%F a(n) = 2*(t+1)+(t*(t+1)/2-n), where t=floor((-1+sqrt(8*n-7))/2). (End)
%F a(n) = (r^2 + 3*r - 2*n)/2, where r = round(sqrt(2*n)). - _Wesley Ivan Hurt_, Sep 19 2021
%F a(n) = A105020(n-1)/A002260(n). - _Wesley Ivan Hurt_, Sep 22 2021
%e First few rows of the triangle are:
%e 1;
%e 3, 2;
%e 5, 4, 3;
%e 7, 6, 5, 4;
%e 9, 8, 7, 6, 5;
%e ...
%p A128076 := proc(n,k)
%p 2*n-k ;
%p end proc:
%p seq(seq( A128076(n,k),k=1..n),n=1..12) ;# _R. J. Mathar_, Sep 27 2021
%t Table[(Round[Sqrt[2 n]]^2 + 3 Round[Sqrt[2 n]] - 2 n)/2, {n, 100}] (* _Wesley Ivan Hurt_, Sep 19 2021 *)
%Y Cf. A128064, A004736, A000326 (row sums), A003056, A002260, A002024, A131914, A209304, A094727 (rows reversed).
%K nonn,tabl,easy
%O 1,2
%A _Gary W. Adamson_, Feb 14 2007
%E NAME simplified. - _R. J. Mathar_, Sep 27 2021
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