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Triangle of numbers a(r,j) = j*(j+1) mod r+2, r>=1, j=1..r.
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%I #31 Jun 30 2023 17:21:53

%S 2,2,2,2,1,2,2,0,0,2,2,6,5,6,2,2,6,4,4,6,2,2,6,3,2,3,6,2,2,6,2,0,0,2,

%T 6,2,2,6,1,9,8,9,1,6,2,2,6,0,8,6,6,8,0,6,2,2,6,12,7,4,3,4,7,12,6,2,2,

%U 6,12,6,2,0,0,2,6,12,6,2,2,6,12,5,0,12,11,12,0,5,12,6,2,2,6,12,4,14,10,8,8,10,14,4,12,6,2

%N Triangle of numbers a(r,j) = j*(j+1) mod r+2, r>=1, j=1..r.

%C Called Dudley Triangle after the American mathematician and writer Underwood Dudley (b. 1937). - _Amiram Eldar_, Jun 10 2021

%C Central terms are A014682(n), n>0. - _Philippe Deléham_, May 11 2023

%D Clifford A. Pickover, The Dudley Triangle", Ch. 59 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford, England: Oxford University Press, 2001, pp. 144-145.

%H Alois P. Heinz, <a href="/A036238/b036238.txt">Rows n = 1..200, flattened</a>

%H Underwood Dudley, <a href="http://www.jstor.org/stable/2690418">Problem 1277: An infinite triangular array</a>, Math. Mag., Vol. 60, No. 5 (1987), p. 328.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dudley_triangle">Dudley triangle</a>.

%e Triangle starts:

%e 2;

%e 2, 2;

%e 2, 1, 2;

%e 2, 0, 0, 2;

%e 2, 6, 5, 6, 2;

%e 2, 6, 4, 4, 6, 2;

%e 2, 6, 3, 2, 3, 6, 2;

%e ...

%t Table[Mod[j (j + 1), r + 2], {r, 14}, {j, r}] // Flatten (* _Michael De Vlieger_, Sep 23 2015 *)

%o (C) #include <stdio.h> #include <stdlib.h> #define MAX_ROWS 100 #define USAGE "Usage: 'A036238 num' where num is the last row of the triangle to compute\n" int main(int argc, char *argv[]) { unsigned long i, j, end, ans; if (argc < 2) { fprintf(stderr, USAGE); return EXIT_FAILURE; } end = atoi(argv[1]); end = (end >= MAX_ROWS) ? MAX_ROWS: end; fprintf(stdout, "Values: "); for (i = 1; i <= end; i++) { for (j = 1; j <= i; j++) { ans = j * (j + 1) % (i +2); fprintf(stdout, "%ld,", ans); } } fprintf(stdout, "\n"); return EXIT_SUCCESS; } /* Larry Reeves (larryr(AT)acm.org), Mar 31 2000 */

%o (PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(k*(k+1) % (n+2), ", ");); print(););} \\ _Michel Marcus_, Sep 23 2015

%Y Cf. A014682.

%K nonn,look,easy,tabl

%O 1,1

%A _N. J. A. Sloane_.

%E More terms from Larry Reeves (larryr(AT)acm.org), Mar 31 2000