Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #40 Nov 04 2019 08:55:24
%S 1,2,3,3,4,0,5,5,6,0,6,7,7,0,8,0,0,9,9,9,10,0,0,10,11,11,0,0,12,0,12,
%T 0,13,13,0,0,14,0,0,14,15,15,15,0,15,16,0,0,0,0,17,17,0,0,0,18,0,18,
%U 18,0,19,19,0,0,0,20,0,0,0,20,21,21,21,0,0,21,22,0,0,22,0,0,23,23,0,0,0,0,24,0,24,0,0,0
%N Triangle read by rows: T(n,k) = n*A237048(n,k).
%C Conjecture: T(n,k) = n, is also the sum of the parts of the partition of n into k consecutive parts, if such a partition exists, otherwise T(n,k) = 0.
%e Triangle begins:
%e 1;
%e 2;
%e 3, 3;
%e 4, 0;
%e 5, 5;
%e 6, 0, 6;
%e 7, 7, 0;
%e 8, 0, 0;
%e 9, 9, 9;
%e 10, 0, 0, 10;
%e 11, 11, 0, 0;
%e 12, 0, 12, 0;
%e 13, 13, 0, 0;
%e 14, 0, 0, 14;
%e 15, 15, 15, 0, 15;
%e 16, 0, 0, 0, 0;
%e 17, 17, 0, 0, 0;
%e 18, 0, 18, 18, 0;
%e 19, 19, 0, 0, 0;
%e 20, 0, 0, 0, 20;
%e 21, 21, 21, 0, 0, 21;
%e 22, 0, 0, 22, 0, 0;
%e 23, 23, 0, 0, 0, 0;
%e 24, 0, 24, 0, 0, 0;
%e 25, 25, 0, 0, 25, 0;
%e 26, 0, 0, 26, 0, 0;
%e 27, 27, 27, 0, 0, 27;
%e 28, 0, 0, 0, 0, 0, 28;
%e ...
%o (PARI) t(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0); \\ A237048
%o tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(n*t(n, k), ", "); ); print(); ); } \\ _Michel Marcus_, Nov 04 2019
%Y Row sums give A245579.
%Y Row n has length A003056(n).
%Y Column k starts in row A000217(k).
%Y The number of positive terms in row n is A001227(n), the number of partitions of n into consecutive parts.
%Y Cf. A196020, A211343, A235791, A236104, A237048, A237591, A237593, A245579, A285900, A285914, A286013.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, May 02 2017