A334462 - OEIS

%I #21 Feb 21 2023 04:24:16

%S 1,1,1,1,1,1,2,1,0,1,2,1,0,1,2,1,0,1,2,1,0,1,2,1,0,3,1,2,0,1,0,0,1,2,

%T 3,1,0,0,1,2,0,1,0,3,1,2,0,1,0,0,1,2,3,1,0,0,1,2,0,1,0,3,1,2,0,4,1,0,

%U 0,0,1,2,3,0,1,0,0,0,1,2,0,4,1,0,3,0,1,2,0,0,1,0,0,0,1,2,3,4,1,0,0,0,1,2,0

%N Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 4, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

%C Since the trivial partition n is counted, so T(n,1) = 1.

%C This is also an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th hexagonal number.

%C This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.

%C For a general theorem about the triangles of this family see A285914.

%F T(n,k) = k*A334460(n,k).

%e Triangle begins (rows 1..28):

%e 1;

%e 1;

%e 1;

%e 1;

%e 1;

%e 1, 2;

%e 1, 0;

%e 1, 2;

%e 1, 0;

%e 1, 2;

%e 1, 0;

%e 1, 2;

%e 1, 0;

%e 1, 2;

%e 1, 0, 3;

%e 1, 2, 0;

%e 1, 0, 0;

%e 1, 2, 3;

%e 1, 0, 0;

%e 1, 2, 0;

%e 1, 0, 3;

%e 1, 2, 0;

%e 1, 0, 0;

%e 1, 2, 3;

%e 1, 0, 0;

%e 1, 2, 0;

%e 1, 0, 3;

%e 1, 2, 0, 4;

%e ...

%e For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. The number of parts of these partitions are 1, 2, 4 respectively, so the 28th row of the triangle is [1, 2, 0, 4].

%Y Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), this sequence (d=4), A334540 (d=5).

%Y Cf. A000384, A334460, A334461.

%K nonn,tabf

%O 1,7

%A _Omar E. Pol_, May 05 2020