A337719 The number of maximally large absolute-difference triangles consisting of positive integers <= n.

1, 2, 4, 8, 16, 32, 44, 72, 128, 220, 380, 620, 1232, 2400, 3988, 7008, 14260, 25512, 50944, 105560, 197880, 381432, 785984, 1443992, 2981200, 6623144, 13044340, 26020924, 55781760, 108592260, 231819360, 526660160, 1071224176, 2231977656, 4950184948, 10009562624

Offset

1,2

Comments

a(17) is the first term that is more than twice its predecessor.

All terms after a(2) are divisible by four. This is because valid starting layers (of length greater than two) produce distinct valid starting layers when subjected to either or both of two transformations.

.

1 1

2 1 1 2

1 3 2 2 3 1

* | *

* | *

* | *

------|------

* | *

* | *

* | *

3 1 2 2 1 3

2 1 1 2

1 1

.

There is the obvious reflection about the y-axis (reversal), and there is the somewhat less obvious reflection about the x-axis. Reflection about the x-axis is valid because absolute differences are maintained. Note that it is not possible for a solution to be equivalent to any of its own transformations. If it were, the base layer or the layer that succeeds it would need to be palindromic. This is invalid because any absolute-difference triangle with a palindromic base and a height greater than one is topped with a zero.

Links

Table of n, a(n) for n=1..36.

Samuel B. Reid, Python program for A337719

Samuel B. Reid, C program for A337719

Example

a(5) = 16
.
  1 2 5 1 2   3 2 5 1 2   3 4 1 5 4   5 4 1 5 4
   1 3 4 1     1 3 4 1     1 3 4 1     1 3 4 1
    2 1 3       2 1 3       2 1 3       2 1 3
     1 2         1 2         1 2         1 2
      1           1           1           1
.
  3 1 5 4 2   3 5 1 2 4   1 4 5 1 4   5 2 1 5 2
   2 4 1 2     2 4 1 2     3 1 4 3     3 1 4 3
    2 3 1       2 3 1       2 3 1       2 3 1
     1 2         1 2         1 2         1 2
      1           1           1           1
.
  2 4 5 1 3   4 2 1 5 3   2 5 1 2 5   4 1 5 4 1
   2 1 4 2     2 1 4 2     3 4 1 3     3 4 1 3
    1 3 2       1 3 2       1 3 2       1 3 2
     2 1         2 1         2 1         2 1
      1           1           1           1
.
  2 1 5 2 1   2 1 5 2 3   4 5 1 4 3   4 5 1 4 5
   1 4 3 1     1 4 3 1     1 4 3 1     1 4 3 1
    3 1 2       3 1 2       3 1 2       3 1 2
     2 1         2 1         2 1         2 1
      1           1           1           1
.

Prog

(Python and C) See Links section.

Crossrefs

Cf. A062684, A094225.

Sequence in context: A371294 A133408 A335019 * A289275 A316758 A316750

Adjacent sequences: A337716 A337717 A337718 * A337720 A337721 A337722

Keyword

nonn

Author

Samuel B. Reid, Sep 16 2020

Status

approved