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%I #72 Jun 24 2022 23:38:04
%S 1,2,2,3,4,4,5,5,6,7,7,8,8,9,9,10,11,11,12,12,13,13,14,14,15,16,16,17,
%T 17,18,18,19,19,20,20,21,22,22,23,23,24,24,25,25,26,26,27,27,28,29,29,
%U 30,30,31,31,32,32,33,33,34,34,35,35,36,37,37,38,38
%N Triangle T(n,k) with rows of length 2n-1 filled with consecutive integers, each appearing twice except for the last term, T(n,2n-1) = n(n+1)/2.
%C This sequence is derived from the water retention heights between a group of hexagonally packed, vertically oriented cylinders. The cylinders fill an equilateral triangle. The cylinder height is specified by sequential natural numbers.
%C This entry is motivated by A258445 and A259052. The water retention on mathematical surfaces concept uses the integer values to specify the height of cylinders. Three touching cylinders define the retention unit and thus a relationship between 3 integers. Taking the minimum, maximum or sum of these three integers provides the data points to construct a new triangle. The water retention between three cylinders is the minimum of the three heights.
%C The row sums for the minimum of three terms in this triangle are A081436.
%C The retention values between rows can be thought of having a wave form with peak and trough values. These peak and trough values correspond to A006002 and A006003. This water retention business shows that A081436(n) = A006002(n) + A006003(n+1).
%C In general any 2D arrangement of numbers can be subdivided by up and down retention triples.
%C The terms of the "flattened" sequence (concatenation of the rows) are the positive integers, repeated twice except for (1, 3, 6, 10, ...) = triangular numbers A000217, which are only listed once. - _M. F. Hasler_, Aug 11 2015
%H Craig Knecht, <a href="/A259549/a259549_3.jpg">Micro and macro equilateral triangles.</a>
%H Craig Knecht, <a href="/A259549/a259549.png">Wave peak and trough.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water retention on mathematical surfaces">Water Retention on Mathematical Surfaces</a>
%F a(n) = (1/4)*(2*t(n) + 1 - (-1)^t(n)), where t(n) = n + floor(sqrt(n)). - _Ridouane Oudra_, Jun 08 2019
%e The irregular triangle T(n,k) starts:
%e n/k 1 2 3 4 5 6 7 8 9 10 11
%e 1: 1
%e 2: 2 2 3
%e 3: 4 4 5 5 6
%e 4: 7 7 8 8 9 9 10
%e 5: 11 11 12 12 13 13 14 14 15
%e 6: 16 16 17 17 18 18 19 19 20 20 21
%o (PARI) a259549(nmax) = { /* Give the first nmax rows */
%o my(L=List(), t);
%o for(n=1, nmax,
%o t=(n^2-n+2)/2;
%o for(k=0, n-2,
%o listput(L, t+k); listput(L, t+k)
%o );
%o listput(L, n*(n+1)/2)
%o );
%o Vec(L)
%o }
%o a259549(6) \\ _Colin Barker_, Jul 04 2015
%o (PARI) A259549(n,k)=(n^2-n+k+1)\2 \\ _M. F. Hasler_, Aug 11 2015
%Y Partial sums of A071028.
%K nonn,tabf
%O 1,2
%A _Craig Knecht_, Jun 30 2015
%E More terms from _Colin Barker_, Jul 04 2015
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