A161983 Irregular triangle read by rows: the group of 2n + 1 integers starting at A014105(n).

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

Offset: 0

Author

Juri-Stepan Gerasimov, Jun 23 2009

Keywords

• nonn
• tabf

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 <= n-1. 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

Examples

			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;
...
		

Links

• Michael Boardman, Proof Without Words: Pythagorean Runs, Math. Mag., 73 (2000), 59.

Crossrefs

Union of A014105 and A317304.

The complement is A162917.

Column 1 gives A014105.

Right border gives A014106.

Row sums give the even-indexed terms of A027480.

Cf. A000290, A014105, A014106, A027480, A162917, A237593, A317304.

Programs

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

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

Extensions

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