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.

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

Offset: 1

Author

Bernard Schott, May 29 2019

Keywords

• nonn
• more

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 2m-1 5 ...

2m 2m-2 2m-4 2m-6 ... 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.

Not to be confused with another definition of trapezoidal numbers, A165513. - N. J. A. Sloane, Jul 13 2019

Examples

			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
		

Links

• "Diophante", A352. Les nombres trapéziens, Sep. 2014 (in French).
• Bert Dobbelaere, C++ program
• Bernard Schott, Examples of numbers in trapezoid

Crossrefs

Cf. A165513.

Extensions

a(25)-a(37) from Bert Dobbelaere, Jul 14 2019