

A337719


The number of maximally large absolutedifference triangles consisting of positive integers <= n.


1



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
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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 yaxis (reversal), and there is the somewhat less obvious reflection about the xaxis. Reflection about the xaxis 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 absolutedifference triangle with a palindromic base and a height greater than one is topped with a zero.


LINKS



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



KEYWORD

nonn


AUTHOR



STATUS

approved



