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Triangle read by rows: T(n, k) = (2*n + 2 - k)*k, for 0 <= k <= n.
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%I #22 Jun 02 2021 15:08:02

%S 0,0,3,0,5,8,0,7,12,15,0,9,16,21,24,0,11,20,27,32,35,0,13,24,33,40,45,

%T 48,0,15,28,39,48,55,60,63,0,17,32,45,56,65,72,77,80,0,19,36,51,64,75,

%U 84,91,96,99,0,21,40,57,72,85,96,105,112,117,120,0,23,44,63,80,95,108,119,128,135,140,143,0,25

%N Triangle read by rows: T(n, k) = (2*n + 2 - k)*k, for 0 <= k <= n.

%C The entries of a row n appear on the diagonal of a square array of dimension n + 1 while filling it with numbers from 0 to n^2 - 1 first along top row and left column, then along 2nd row and 2nd column, 3rd row and 3rd column etc. up to the (single) entry in the n-th row and n-th column. [This may be the preferred order if a set of matrices M is built with requirements on the product M*M.] This vaguely is an alternative to the boustrophedonic re-arrangement of a finite array.

%C The triangle may also be generated by reading half of each second antidiagonal of the array A003991.

%C The numbers appear in reverse order as the numerators in the triangle A061035 before they are reduced with the denominators by cancellation of common factors. - _Paul Curtz_, May 03 2013

%e Triangle starts:

%e [0] 0;

%e [1] 0, 3;

%e [2] 0, 5, 8;

%e [3] 0, 7, 12, 15;

%e [4] 0, 9, 16, 21, 24;

%e [5] 0, 11, 20, 27, 32, 35;

%e [6] 0, 13, 24, 33, 40, 45, 48;

%e [7] 0, 15, 28, 39, 48, 55, 60, 63;

%e [8] 0, 17, 32, 45, 56, 65, 72, 77, 80;

%e [9] 0, 19, 36, 51, 64, 75, 84, 91, 96, 99.

%e .

%e The row n = 3, for example, is created by reading the 4 X 4 square array downwards its main diagonal.

%e 0, 1, 3, 5;

%e 2, 7, 8, 10;

%e 4, 9, 12, 13;

%e 6, 11, 14, 15;

%p T := proc(n, k) option remember; if k = 0 then 0 elif k = 1 then 2*n+1 else

%p T(n, k-1) + T(n-k+1, 1) fi end:

%p for n from 0 to 9 do seq(T(n, k), k=0..n) od; # _Peter Luschny_, Jun 02 2021

%Y Cf. A016061 (row sums), A045944 (central), A005563 (main diagonal).

%Y Cf. A005408 (column k=2), A008586 (column k=3), A016945 (column k=4).

%Y Cf. A003991, A061035.

%K easy,nonn,tabl

%O 0,3

%A _R. J. Mathar_, May 03 2013

%E Offset set to 0 and edited by _Peter Luschny_, Jun 02 2021