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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. 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 (list; graph; refs; listen; history; text; internal format)
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        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

LINKS

Table of n, a(n) for n=1..37.

"Diophante", A352. Les nombres trapéziens, Sep. 2014 (in French).

Bert Dobbelaere, C++ program

Bernard Schott, Examples of numbers in trapezoid

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

Cf. A165513.

Sequence in context: A138591 A136492 A062506 * A213199 A184987 A242437

Adjacent sequences:  A308465 A308466 A308467 * A308469 A308470 A308471

KEYWORD

nonn,more

AUTHOR

Bernard Schott, May 29 2019

EXTENSIONS

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

STATUS

approved

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Last modified September 24 15:40 EDT 2021. Contains 347643 sequences. (Running on oeis4.)