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%I #47 Sep 07 2021 04:10:33
%S 0,3,4,5,10,11,12,13,14,21,22,23,24,25,26,27,36,37,38,39,40,41,42,43,
%T 44,55,56,57,58,59,60,61,62,63,64,65,78,79,80,81,82,83,84,85,86,87,88,
%U 89,90,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,136,137
%N Irregular triangle read by rows: the group of 2n + 1 integers starting at A014105(n).
%C 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.
%C 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
%C 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
%H Michael Boardman, <a href="http://www.jstor.org/stable/2691496">Proof Without Words: Pythagorean Runs</a>, Math. Mag., 73 (2000), 59.
%F As a triangle, T(n,k) = 2n^2 + 2n + k where -n <= k <= n and n = 0,1,... - _Dennis P. Walsh_, Aug 09 2013
%F 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
%e Triangle begins:
%e 0;
%e 3, 4, 5;
%e 10, 11, 12, 13, 14;
%e 21, 22, 23, 24, 25, 26, 27;
%e 36, 37, 38, 39, 40, 41, 42, 43, 44;
%e 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65;
%e 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90;
%e ...
%p seq(seq(2*n^2+2*n+k,k=-n..n),n=0..10); # _Dennis P. Walsh_, Aug 09 2013
%p seq(n+floor(sqrt(n))*(floor(sqrt(n))+1),n=0..100); # _Dennis P. Walsh_, Aug 09 2013
%Y Union of A014105 and A317304.
%Y The complement is A162917.
%Y Column 1 gives A014105.
%Y Right border gives A014106.
%Y Row sums give the even-indexed terms of A027480.
%Y Cf. A000290, A014105, A014106, A027480, A162917, A237593, A317304.
%K nonn,tabf
%O 0,2
%A _Juri-Stepan Gerasimov_, Jun 23 2009
%E Definition clarified, 8th row terms corrected by _R. J. Mathar_, Jul 19 2009
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