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%I #48 Jul 14 2019 07:31:07
%S 3,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,23,24,25,27,29,30,31,33,
%T 35,36,37,39,41,42,43,45,47,48,49,51,53
%N "Trapezoidal numbers": numbers k such that the integers from 1 to k can be arranged in a trapezoid of H lines containing respectively L, L-1, L-2, ..., L-H+1 numbers from top to bottom. The rule is that from the second line, each integer is equal to the absolute value of the difference between the two numbers above it.
%C These numbers are called "nombres trapéziens" in French.
%C Some results from the article by "Diophante" (problème A352):
%C The powers of 2 are not trapezoidal.
%C Every odd number >= 3 is trapezoidal. In the case of k = 2m+1, a pattern can always be obtained with a trapezoid of height H = 2. The first line has the m+1 odd integers and the second the m even integers decreasing from 2m to 2, with this following arrangement:
%C 1 2m+1 3 2m-1 5 ...
%C 2m 2m-2 2m-4 2m-6 ... 2
%C If H = L, the trapezoid becomes a triangle (examples for 3, 6 and 10 that are triangular numbers but 28 is not in trapezoid).
%C When an integer is trapezoidal, the number of ways for this to happen varies greatly; up to 30, the number of distinct solutions is greater when k is multiple of 6. Two symmetric trapezoids are considered to be identical.
%C It is not known if this sequence has a finite number of even terms.
%C If 34 is trapezoidal then the only possible trapezoid is necessarily of the form L = 10 and H = 4, and,
%C if 36 is trapezoidal, there are only two possible trapezoid forms, the first has L = 8 and H = 8 (it is a triangle) and the second one has L = 13 and H = 3.
%C Not to be confused with another definition of trapezoidal numbers, A165513. - _N. J. A. Sloane_, Jul 13 2019
%H "Diophante", <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/2924-2014-06-28-20-45-29">A352. Les nombres trapéziens</a>, Sep. 2014 (in French).
%H Bert Dobbelaere, <a href="/A308468/a308468.cpp.txt">C++ program</a>
%H Bernard Schott, <a href="/A308468/a308468.pdf">Examples of numbers in trapezoid</a>
%e for k = 9: 1 9 3 7 5
%e 8 6 4 2
%e ------------------------------------------------------
%e for k = 10: 8 1 10 6
%e 7 9 4
%e 2 5
%e 3
%Y Cf. A165513.
%K nonn,more
%O 1,1
%A _Bernard Schott_, May 29 2019
%E a(25)-a(37) from _Bert Dobbelaere_, Jul 14 2019
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