# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a334460
Showing 1-1 of 1

%I A334460 #22 Oct 02 2020 09:45:47
%S A334460 1,1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,
%T A334460 1,1,0,0,1,1,0,1,0,1,1,1,0,1,0,0,1,1,1,1,0,0,1,1,0,1,0,1,1,1,0,1,1,0,
%U A334460 0,0,1,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0,1,1,1,1,1,0,0,0,1,1,0,0
%N A334460 Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by 4, and the first element of column k is in the row that is the k-th hexagonal number (A000384).
%C A334460 T(n,k) is 0 or 1, so T(n,k) represents the "existence" of the mentioned partition: 1 = exists, 0 = does not exist.
%C A334460 Since the trivial partition n is counted, so T(n,1) = 1.
%C A334460 This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists 1'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 A334460 This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.
%C A334460 For a general theorem about the triangles of this family see A303300.
%e A334460 Triangle begins (rows 1..28):
%e A334460 1;
%e A334460 1;
%e A334460 1;
%e A334460 1;
%e A334460 1;
%e A334460 1, 1;
%e A334460 1, 0;
%e A334460 1, 1;
%e A334460 1, 0;
%e A334460 1, 1;
%e A334460 1, 0;
%e A334460 1, 1;
%e A334460 1, 0;
%e A334460 1, 1;
%e A334460 1, 0, 1;
%e A334460 1, 1, 0;
%e A334460 1, 0, 0;
%e A334460 1, 1, 1;
%e A334460 1, 0, 0;
%e A334460 1, 1, 0;
%e A334460 1, 0, 1;
%e A334460 1, 1, 0;
%e A334460 1, 0, 0;
%e A334460 1, 1, 1;
%e A334460 1, 0, 0;
%e A334460 1, 1, 0;
%e A334460 1, 0, 1;
%e A334460 1, 1, 0, 1;
%e A334460 ...
%e A334460 For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a partition. They are [28], [16, 12] and [13, 9, 5, 1]. The number of parts of these partitions are 1, 2, 4 respectively. There are no partitions of this kind with three parts, so the 28th row of the triangle is [1, 1, 0, 1].
%p A334460 A334460 := proc(n, k)
%p A334460     local first1 ;
%p A334460     first1 := A000384(k) ;
%p A334460     if n < first1 then
%p A334460         0 ;
%p A334460     elif modp(n-first1, k) = 0 then
%p A334460         1;
%p A334460     else
%p A334460         0;
%p A334460     end if;
%p A334460 end proc:
%p A334460 for n from 1 to 40 do
%p A334460     for k from 1 do
%p A334460         if n>= A000384(k) then
%p A334460             printf("%d, ", A334460(n, k)) ;
%p A334460         else
%p A334460             break;
%p A334460         end if;
%p A334460     end do:
%p A334460     printf("\n") ;
%p A334460 end do: # _R. J. Mathar_, Oct 02 2020
%Y A334460 Row sums give A334461.
%Y A334460 Triangles of the same family where the parts differ by d are A051731 (d=0), A237048 (d=1), A303300 (d=2), A330887 (d=3), this sequence (d=4).
%Y A334460 Cf. A000384, A327262, A334462.
%K A334460 nonn,tabf
%O A334460 1
%A A334460 _Omar E. Pol_, May 01 2020

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE