Definition
Two theoretical symmetric patterns of length k1 and k2
(k1 < k2) are called successive if the pattern of length k1 is a symmetrically located subpattern of the pattern of length k2.

Example

pattern of length 17

0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

pattern of length 19

0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

Symmetrically located subpattern of the pattern of length 19

6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246

Normalize this subpattern and get the following pattern of length 17

0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

Thus, the patterns shown are successive.

That is why from the found symmetrical 19-tuplet with a minimum diameter of 252

9425346484752129657862217: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

we get a symmetrical 17-tuplet with a minimum diameter of 240

9425346484752129657862223: 0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

If a symmetrical 19-tuplet with a minimum diameter of 252 is found, then a symmetrical 17-tuplet with a minimum diameter of 240 is automatically found.
The reverse is not true.

Example

1006882292528806742267: 0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

This is a symmetric 17-tuplet with a minimum diameter of 240 and a successive pattern (relative to the symmetric 19-tuplet).
However, this tuple does not continue to a symmetric 19-tuplet with a minimum diameter of 252.

Conclusion: the existence of a symmetric 17-tuplet with a minimum diameter of 240 and a successive pattern is a necessary but not sufficient condition for the existence of a symmetric 19-tuplet with a minimum diameter of 252.

We should find this symmetric 17-tuplet in the BOINC project ODLK2025

9425346484752129657862223: 0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

because it is in the search range of this project.

Therefore, this symmetrical 19-tuplet with a minimum diameter of 252

9425346484752129657862217: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

we will also find.

We need to find many symmetrical 17-tuplets with a minimum diameter of 240 and with a successive pattern in the project in order to find several symmetrical 19-tuplets with a minimum diameter of 252.

Next, the pattern of the symmetrical 19-tuplet with a minimum diameter of 252 is successive with the pattern of the symmetrical 21-tuplet with a diameter of 360

0, 54, 60, 66, 84, 96, 126, 144, 150, 174, 180, 186, 210, 216, 234, 264, 276, 294, 300, 306, 360

Therefore, some 19-tuplets with a minimum diameter of 252 can continue to a symmetrical 21-tuplet with a diameter of 360.
To do this, we need to find many 19-tuplets with a minimum diameter of 252.

And that's not all!
I found a series of successive patterns for symmetric k-tuplets for k=17 – 27 (with a step of 2).

See out this interesting series

symmetrical 17-tuplet with a minimum diameter of 240

0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

-->
symmetrical 19-tuplet with a minimum diameter of 252

0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

-->
symmetrical 21-tuplet with a diameter of 360

0, 54, 60, 66, 84, 96, 126, 144, 150, 174, 180, 186, 210, 216, 234, 264, 276, 294, 300, 306, 360

-->
symmetrical 23-tuplet with a diameter of 408

0, 24, 78, 84, 90, 108, 120, 150, 168, 174, 198, 204, 210, 234, 240, 258, 288,300, 318, 324, 330, 384, 408

-->
symmetrical 25-tuplet with a minimum diameter of 420

0, 6, 30, 84, 90, 96, 114, 126, 156, 174, 180, 204, 210, 216, 240, 246, 264, 294, 306, 324, 330, 336, 390, 414, 420

-->
symmetrical 27-tuplet with a minimum diameter of 432

0, 6, 12, 36, 90, 96, 102, 120, 132, 162, 180, 186, 210, 216, 222, 246, 252, 270, 300, 312, 330, 336, 342, 396, 420, 426, 432

This symmetrical 19-tuplet with a minimum diameter of 252

9425346484752129657862217: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

is represented in detail as

{9425346484752129657862217, 9425346484752129657862223, 9425346484752129657862229, 9425346484752129657862247,
9425346484752129657862259, 9425346484752129657862289, 9425346484752129657862307, 9425346484752129657862313,
9425346484752129657862337, 9425346484752129657862343, 9425346484752129657862349, 9425346484752129657862373,
9425346484752129657862379, 9425346484752129657862397, 9425346484752129657862427, 9425346484752129657862439,
9425346484752129657862457, 9425346484752129657862463, 9425346484752129657862469}

Now I'll show you the matryoshka dolls

{9425346484752129657862217, 9425346484752129657862223, 9425346484752129657862229, 9425346484752129657862247,
9425346484752129657862259, 9425346484752129657862289, 9425346484752129657862307, 9425346484752129657862313,
9425346484752129657862337, 9425346484752129657862343, 9425346484752129657862349, 9425346484752129657862373,
9425346484752129657862379, 9425346484752129657862397, 9425346484752129657862427, 9425346484752129657862439,
9425346484752129657862457, 9425346484752129657862463, 9425346484752129657862469}

{9425346484752129657862217, 9425346484752129657862223, 9425346484752129657862229, 9425346484752129657862247,
9425346484752129657862259, 9425346484752129657862289, 9425346484752129657862307, 9425346484752129657862313,
9425346484752129657862337, 9425346484752129657862343, 9425346484752129657862349, 9425346484752129657862373,
9425346484752129657862379, 9425346484752129657862397, 9425346484752129657862427, 9425346484752129657862439,
9425346484752129657862457, 9425346484752129657862463, 9425346484752129657862469}

{9425346484752129657862217, 9425346484752129657862223, 9425346484752129657862229, 9425346484752129657862247,
9425346484752129657862259, 9425346484752129657862289, 9425346484752129657862307, 9425346484752129657862313,
9425346484752129657862337, 9425346484752129657862343, 9425346484752129657862349, 9425346484752129657862373,
9425346484752129657862379, 9425346484752129657862397, 9425346484752129657862427, 9425346484752129657862439,
9425346484752129657862457, 9425346484752129657862463, 9425346484752129657862469}

and so on

Dear project participants!

If you are interested in tuple theory and want to understand everything, please ask your questions.
I will be happy to answer you.

Some people write that the goal of the BOINC project ODLK2025 is not clear.
I am trying to explain the goal, strategy and tactics of the search.

Unfortunately, this symmetrical 19-tuplet with a minimum diameter of 252

9425346484752129657862217: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

does not continue to 21-tuplet with a diameter of 360.

No luck so far.

We will continue to search for 19-tuplets with a minimum diameter of 252.