

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
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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

Table of n, a(n) for n=0..65.
Michael Boardman, Proof Without Words: Pythagorean Runs, Math. Mag., 73 (2000), 59.


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
As sequence, a(n) = n + floor(sqrt(n))*(floor(sqrt(n)) + 1); equivalently, a(n) = n + A000196(n)*(A000196(n)+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(seq(2*n^2+2*n+k, k=n..n), n=0..10); # Dennis P. Walsh, Aug 09 2013
seq(n+floor(sqrt(n))*(floor(sqrt(n))+1), n=0..100); # Dennis P. Walsh, Aug 09 2013


CROSSREFS

Union of A014105 and A317304.
The complement is A162917.
Column 1 gives A014105.
Right border gives A014106.
Row sums give the evenindexed terms of A027480.
Cf. A000290, A014105, A014106, A027480, A162917, A237593, A317304.
Sequence in context: A086186 A014463 A340015 * A047364 A274519 A139445
Adjacent sequences: A161980 A161981 A161982 * A161984 A161985 A161986


KEYWORD

nonn,tabf


AUTHOR

JuriStepan Gerasimov, Jun 23 2009


EXTENSIONS

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


STATUS

approved



