Step by Step solution about How to resolve the algorithm Van Eck sequence step by step in the Mathematica/Wolfram Language programming language

Explanation:

This code generates a list of sublists, where each sublist consists of alternating non-negative integers and negative integers. The integers are generated using a nested iteration process.

Details:

1. TakeList[Nest[..., {10, -10}], {10}]:

   • TakeList[Nest[...], {10, -10}] applies the Nest function to the initial value {0} and takes the first 10 elements of the result.
   • Nest[... {0}, 999] repeatedly applies the nested function to {0} a total of 999 times.

2. Nest[If[MemberQ[..., #//Last]], Join[#, Length[#] - Last@Position[..., #//Last]], Append[#, 0]]&, {0}, 999]

   • This is the nested function that generates the sequence of sublists.
   • #//Most and #//Last extract the first and last elements of the current sublist, respectively.
   • If[MemberQ[..., #//Last], ...]: This condition checks if the last element of the current sublist exists in the first elements (i.e., Most) of the sublist.
   • If the condition is true, it joins the current sublist with the index of the last element minus the length of the sublist (Join[#, Length[#] - Last@Position[..., #//Last]]). This effectively adds a new non-negative integer to the sublist.
   • If the condition is false, it appends a 0 to the current sublist.

3. Column: This function formats the output as a column, with each sublist on a separate line.

Result:

The output of the code is a list of 10 sublists, each containing alternating non-negative integers and negative integers. For example:

{{0}, {0, -1}, {0, -1, 0}, {0, -1, 0, -2}, {0, -1, 0, -2, 0}, {0, -1, 0, -2, 0, -3}, {0, -1, 0, -2, 0, -3, 0}, {0, -1, 0, -2, 0, -3, 0, -4}, {0, -1, 0, -2, 0, -3, 0, -4, 0}, {0, -1, 0, -2, 0, -3, 0, -4, 0, -5}}

TakeList[Nest[If[MemberQ[#//Most, #//Last], Join[#, Length[#] - Last@Position[#//Most, #//Last]], Append[#, 0]]&, {0}, 999], {10, -10}] // Column