

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, L1, L2, ..., LH+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.


1



3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53
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OFFSET

1,1


COMMENTS

These numbers are called "nombres trapéziens" in French.
Some results from the article by "Diophante" (problème A352):
The powers of 2 are not trapezoidal.
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:
1 2m+1 3 2m1 5 ...
2m 2m2 2m4 2m6 ... 2
If H = L, the trapezoid becomes a triangle (examples for 3, 6 and 10 that are triangular numbers but 28 is not in trapezoid).
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.
It is not known if this sequence has a finite number of even terms.
If 34 is trapezoidal then the only possible trapezoid is necessarily of the form L = 10 and H = 4, and,
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.


LINKS



EXAMPLE

for k = 9: 1 9 3 7 5
8 6 4 2

for k = 10: 8 1 10 6
7 9 4
2 5
3


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



