login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

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 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286013 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the positive integers starting with k, interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2. 16
1, 2, 3, 2, 4, 0, 5, 3, 6, 0, 3, 7, 4, 0, 8, 0, 0, 9, 5, 4, 10, 0, 0, 4, 11, 6, 0, 0, 12, 0, 5, 0, 13, 7, 0, 0, 14, 0, 0, 5, 15, 8, 6, 0, 5, 16, 0, 0, 0, 0, 17, 9, 0, 0, 0, 18, 0, 7, 6, 0, 19, 10, 0, 0, 0, 20, 0, 0, 0, 6, 21, 11, 8, 0, 0, 6, 22, 0, 0, 7, 0, 0, 23, 12, 0, 0, 0, 0, 24, 0, 9, 0, 0, 0, 25, 13, 0, 0, 7, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture 1: T(n,k) is the largest part of the partition of n into k consecutive parts, if T(n,k) > 0.

Conjecture 2: row sums give A286015.

Trapezoidal interpretation from Peter Munn, Jun 18 2017: (Start)

There is one to one correspondence between nonzero T(n,k) and trapezoidal area patterns of n dots on a triangular grid, if we include the limiting cases of triangular patterns, straight lines (k=1) or a single dot (k=n=1). The corresponding pattern has T(n,k) dots in its longest side, k dots in the two adjacent sides and T(n,k)-k+1 dots in the fourth side (where a count of 1 dot may be understood as signifying that side's absence).

Reason: From the definition, for k >= 1, m >= 0, T(A000217(k)+km,k) = k+m, where A000217(k) = k(k+1)/2, the k-th triangular number. First element of column k is T(A000217(k),k) = k: this matches a triangular pattern of A000217(k) dots with 3 sides of k dots. Looking at this pattern as k rows of 1..k dots, extend each row by m dots to create a trapezoidal pattern of A000217(k)+km dots with a longest side of k+m dots and adjacent sides of k dots: this matches T(A000217(k)+km,k) = k+m. As nonzero elements in column k occur at intervals of k, every nonzero T(n,k) has a match. Every trapezoidal pattern can be produced by extending a triangular pattern as described, so they all have a match.

The truth of conjecture 1 follows, since each nonzero T(n,k) = k+m corresponds to a trapezoidal pattern of n dots having k rows with lengths (1+m)..(k+m).

The A270877 sieve is related to this sequence because it eliminates n if it is the sum of consecutive numbers whose largest term has survived the sifting (which may likewise be seen in terms of a trapezoidal dot pattern and its longest side). So the sieve eliminates n if any lesser numbers in A270877 are in row n of this sequence.

(End)

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10944 (Rows 1 <= n <= 528, 528 being first row with 32 columns).

FORMULA

For k >= 1, m >= 0, T(A000217(k)+km,k) = k+m. - Peter Munn, Jun 19 2017

EXAMPLE

Triangle begins:

1;

2;

3, 2;

4, 0;

5, 3;

6, 0, 3;

7, 4, 0;

8, 0, 0;

9, 5, 4;

10, 0, 0, 4;

11, 6, 0, 0;

12, 0, 5, 0;

13, 7, 0, 0;

14, 0, 0, 5;

15, 8, 6, 0, 5;

16, 0, 0, 0, 0;

17, 9, 0, 0, 0;

18, 0, 7, 6, 0;

19, 10, 0, 0, 0;

20, 0, 0, 0, 6;

21, 11, 8, 0, 0, 6;

22, 0, 0, 7, 0, 0;

23, 12, 0, 0, 0, 0;

24, 0, 9, 0, 0, 0;

25, 13, 0, 0, 7, 0;

26, 0, 0, 8, 0, 0;

27, 14, 10, 0, 0, 7;

28, 0, 0, 0, 0, 0, 7;

...

In accordance with the conjecture, for n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The largest parts are 15, 8, 6, 5, respectively, so the 15th row of the triangle is [15, 8, 6, 0, 5].

MATHEMATICA

With[{n = 7}, DeleteCases[#, m_ /; m < 0] & /@ Transpose@ Table[Apply[Join @@ {ConstantArray[-1, #2 - 1], Array[(k + #/k) Boole[Mod[#, k] == 0] &, #1 - #2 + 1, 0]} &, # (# + 1)/2 & /@ {n, k}], {k, n}]] // Flatten (* Michael De Vlieger, Jul 21 2017 *)

CROSSREFS

Row n has length A003056(n).

Column k starts in row A000217(k).

The number of positive terms in row n is A001227(n), the number of partitions of n into consecutive parts.

The last positive term in row n is in column A109814(n).

Cf. A196020, A204217, A211343, A235791, A237048, A237591, A237593, A245579, A270877, A286014, A286015.

Sequence in context: A299115 A179590 A167504 * A208083 A181515 A183045

Adjacent sequences: A286010 A286011 A286012 * A286014 A286015 A286016

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Apr 30 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 06:35 EST 2022. Contains 358582 sequences. (Running on oeis4.)