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[i, k], {i, 0, t - 1}, {j, stot}, {k, j + 1, stot}], Table[Apply[Or, Table[ [i, j], {j, n + i, Max[0, n - i], -2}]], {i, 0, t}], Table[! … [i, k], {i, 0, t}, {j, n + i, Max[0, n - i], -2}, {k, j + 2, n + i}], Table[Apply[Or, Table[ [i, j, k], {k, 0, ktot - 1}]], {i, 0, t - 1}, {j, Max[1, n - i], n + i}], Table[! … [i, j, m], {i, 0, t - 1}, {j, Max[1, n - i], n + i}, {k, 0, ktot - 1}, {m, k + 1, ktot - 1}], [0, s], Cases[MapIndexed[ [Abs[n - First[#2]], First[#2], #1]&, a], [x_, _, _] /; x < t], Table[ [Abs[n - i], i, 0], {i, Length[a] + 1, n + t - 1}], Table[!

Tables of primes
No explicit tables of primes appear to have survived from antiquity, but it seems likely that all primes up to somewhere between 5000 and 10000 were known. … And by the mid-1600s there were printed tables of primes up to 100,000, containing as much data as in plots (c) and (d). In the 1700s and 1800s many tables of number factorizations were constructed; by the 1770s there was a table up to 2 million, and by the 1860s up to 100 million.

MapIndexed[ #1 First[#2] &, Union[Map[# 〚 1, 1 〛 &, #]]] &[ With[{b = Ceiling[Log[2, k]] - 1}, Flatten[Table[ {Table[{Table[{{m, i, n, d}, c} {{m, Mod[i, 2 n - 1 ], n - 1, d}, Quotient[i, 2 n - 1 ], 1}, {n, 2, b}, {i, 0, 2 n - 1}], Table[{ {m, i, 1, d}, c} {{m, -1, 1, d}, i, d}, {i, 0, 1}], Table[ {{m, -1, n, d}, c} {{m, -1, n + 1, d}, c, d}, {n, b - 1}], {{m, -1, b, d}, c} {{0, 0, m}, c, d}}, {d, -1, 1, 2}], Table[{{i, n, m}, c} {{ i + 2 n c, n + 1, m}, c, -1}, {n, 0, b - 1}, {i, 0, 2 n - 1}], With[{r = 2 b }, Table[ If[i + r c ≥ k, {}, Cases[rule, ({m, i + r c} {x_, y_, z_}) {{i, b, m}, c} {{x, Mod[y, r], b, z}, Quotient[y, r], 1})]], {i, 0, r - 1}]]}, {m, s}, {c, 0, 1}]]]]
Some of these states are usually unnecessary, and in the main text such states have been pruned.

Difference tables and polynomials
A common mathematical approach to analyzing sequences is to form a difference table by repeatedly evaluating d[list_] := Drop[list, 1] - Drop[list, -1] . … If the differences are computed modulo k then the difference table corresponds essentially to the evolution of an additive cellular automaton (see page 597 ).

({i_, u_} {j_, v_, r_}) {Map[#[i] {#[i, 1], #[i, 0]} &, {a, b, c, d}], If[r 1, {a[i, u] {a[j], a[j]}, b[i, u] Table[b[j], {4}], c[i, u] Flatten[{Table[b[j], {2v}], Table[c[j], {2 - u}]}], d[i, u] {d[j]}}, {a[i, u] Table[a[j], {2 - u}], b[i, u] {b[j]}, c[i, u] Flatten[{c[j], c[j], Table[d[j], {2v}]}], d[i, u] Table[d[j], {4}]}]}]]}
A Turing machine in state i with a blank tape corresponds to initial condition {a[i], a[i], c[i]} for the tag system.

Common framework [for cellular automaton rules]
The Mathematica built-in function CellularAutomaton discussed on page 867 handles general and totalistic rules in the same framework by using ListConvolve[w, a, r + 1] and taking the weights w to be respectively k^Table[i - 1, {i, 2r + 1}] and Table[1, {2r + 1}] .

If one has an operator whose values are given by some finite table then it is always straightforward to determine whether expressions are equivalent. … And one advantage of this approach is that at least in principle it allows one to handle operators—like those found in many areas of mathematics—that are not based on finite tables. But even for operators given by finite tables it is often difficult to find axiom systems that can successfully reproduce all the results for a particular operator.

{ds[r_, s_] d[r, adrs 〚 s 〛 ], dr[r_, j_] d[r, j + First[#2]]} &, Flatten[segs]]]
seg[i[r_], {a_}] := With[{p = Prime[r]}, Flatten[{Table[i[2], {p}], dr[1, -p], i[1], dr[2, -1], Table[dr[1, 1], {p + 1}]}]]
seg[d[r_, n_], {a_}] := With[{p = Prime[r]}, Flatten[{i[2], dr[ 1, 5], i[1], dr[2, -1], dr[1, 1], ds[1, n], Table[{If[m p - 1, ds[1, a], dr[1, 3p + 2 - m]], Table[i[1], {p}], dr[2, -p], Table[dr[1, 1], {2p - m - 1}], ds[1, a + 1]}, {m, p - 1}]}]]
The initial conditions for the 2-register machine are given by {1, {RMEncode[list], 0}} and the results corresponding to each step in the evolution of the multiregister machine appear whenever register 2 in the 2-register machine is incremented from 0.

Difference tables
See page 1091 .

Rule 60 essentially constructs a difference table for the sequence of elements.