This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A308468 #48 Jul 14 2019 07:31:07 %S A308468 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 A308468 35,36,37,39,41,42,43,45,47,48,49,51,53 %N A308468 "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 A308468 These numbers are called "nombres trapéziens" in French. %C A308468 Some results from the article by "Diophante" (problème A352): %C A308468 The powers of 2 are not trapezoidal. %C A308468 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 A308468 1 2m+1 3 2m-1 5 ... %C A308468 2m 2m-2 2m-4 2m-6 ... 2 %C A308468 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 A308468 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 A308468 It is not known if this sequence has a finite number of even terms. %C A308468 If 34 is trapezoidal then the only possible trapezoid is necessarily of the form L = 10 and H = 4, and, %C A308468 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 A308468 Not to be confused with another definition of trapezoidal numbers, A165513. - _N. J. A. Sloane_, Jul 13 2019 %H A308468 "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 A308468 Bert Dobbelaere, <a href="/A308468/a308468.cpp.txt">C++ program</a> %H A308468 Bernard Schott, <a href="/A308468/a308468.pdf">Examples of numbers in trapezoid</a> %e A308468 for k = 9: 1 9 3 7 5 %e A308468 8 6 4 2 %e A308468 ------------------------------------------------------ %e A308468 for k = 10: 8 1 10 6 %e A308468 7 9 4 %e A308468 2 5 %e A308468 3 %Y A308468 Cf. A165513. %K A308468 nonn,more %O A308468 1,1 %A A308468 _Bernard Schott_, May 29 2019 %E A308468 a(25)-a(37) from _Bert Dobbelaere_, Jul 14 2019