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A301853 Triangle read by rows: T(n,k) gives the number of distinct distances on an n X k pegboard, with n >= 1, 1 <= k <= n.

%I A301853 #39 Nov 01 2021 03:13:35
%S A301853 1,2,3,3,5,6,4,7,9,10,5,9,12,14,15,6,11,15,17,19,20,7,13,18,21,24,26,
%T A301853 27,8,15,21,25,29,31,33,34,9,17,24,29,33,36,39,41,42,10,19,27,33,38,
%U A301853 42,45,48,50,51,11,21,30,37,43,48,51,55,58,60,61,12,23,33,41,48,53,57,61,65,68,70,71
%N A301853 Triangle read by rows: T(n,k) gives the number of distinct distances on an n X k pegboard, with n >= 1, 1 <= k <= n.
%C A301853 Is k*(2*n - k + 1)/2 an upper bound on T(n, k)? - _David A. Corneth_, Mar 28 2018
%H A301853 Peter Kagey, <a href="/A301853/b301853.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%e A301853 Triangle begins:
%e A301853   1;
%e A301853   2,  3;
%e A301853   3,  5,  6;
%e A301853   4,  7,  9, 10;
%e A301853   5,  9, 12, 14, 15;
%e A301853   6, 11, 15, 17, 19, 20;
%e A301853   7, 13, 18, 21, 24, 26, 27;
%e A301853   8, 15, 21, 25, 29, 31, 33, 34;
%e A301853   9, 17, 24, 29, 33, 36, 39, 41, 42;
%e A301853   ...
%o A301853 (PARI) T(n, k) = {my(d=[]); for (i=1, n, for (j=1, k, d = concat(d, (i-1)^2 + (j-1)^2););); #vecsort(d,,8);} \\ _Michel Marcus_, Mar 29 2018
%Y A301853 Cf. A047800, A225273, A301851.
%K A301853 nonn,tabl
%O A301853 1,2
%A A301853 _Peter Kagey_, Mar 27 2018