

A161983


Irregular triangle read by rows: the group of 2n + 1 integers starting at A014105(n).


5



0, 3, 4, 5, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 27, 36, 37, 38, 39, 40, 41, 42, 43, 44, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 136, 137
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

The squares of numbers in each row can be gathered in an equation with the first n terms on one side, the next n+1 terms on the other. The third row, for example, could be rendered as 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
This sequence contains all nonnegative integers that are within a distance of n from 2n^2 + 2n where n is any nonnegative integer. The nonnegative integers that are not in this sequence are of the form 2n^2 + k where n is any positive integer and n <= k <= n1. Also, when n is the product of two consecutive integers, a(n) = 2n; for example, a(20) = 40. See explicit formulas for the sequence in the formula section below.  Dennis P. Walsh, Aug 09 2013
Numbers k with the property that the largest Dyck path of the symmetric representation of sigma(k) has a central valley, n > 0. (Cf. A237593.)  Omar E. Pol, Aug 28 2018


LINKS



FORMULA

As a triangle, T(n,k) = 2n^2 + 2n + k where n <= k <= n and n = 0,1,...  Dennis P. Walsh, Aug 09 2013


EXAMPLE

Triangle begins:
0;
3, 4, 5;
10, 11, 12, 13, 14;
21, 22, 23, 24, 25, 26, 27;
36, 37, 38, 39, 40, 41, 42, 43, 44;
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65;
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90;
...


MAPLE

seq(n+floor(sqrt(n))*(floor(sqrt(n))+1), n=0..100); # Dennis P. Walsh, Aug 09 2013


CROSSREFS

Row sums give the evenindexed terms of A027480.


KEYWORD

nonn,tabf


AUTHOR



EXTENSIONS

Definition clarified, 8th row terms corrected by R. J. Mathar, Jul 19 2009


STATUS

approved



