It contains a total of Round[2 t /3] black elements, and if the last element is dropped, it forms a palindrome.

I have always tried to read original writings—for I have often found that later characterizations drop elements crucial for my purposes, or recast history to simplify pedagogy.

The transitions between these states have probabilities given by m[Map[Length, list]] where m[s_] := With[{q = FoldList[Plus, 0, s]}, ReplacePart[ RotateRight[IdentityMatrix[Last[q]], {0, 1}], 1/Length[s], Flatten[Outer[List, Rest[q], Drop[q, -1] + 1], 1]]] The average spectrum of sequences generated according to these probabilities can be obtained by computing the correlation function for elements a distance r apart ξ [list_, r_] := With[{w = (# - Apply[Plus, #]/Length[#] &)[ Flatten[list]]}, w .

Dropping this 1 yields the pattern below.

TMToRM[rules_] := Module[{segs, adrs}, segs = Map[TMCompile, rules] ; adrs = Thread[Map[First, rules]  Drop[FoldList[Plus, 1, Map[Length, segs]], -1]]; MapIndexed[(# /.

After a few rules with long periods, the periods obtained drop off rapidly.

But if IntegerDigits[x, 2] involves no consecutive 0's then for example f[x] can be obtained from 2^(b[Join[{1, 1}, #], Length[#]] &)[IntegerDigits[x, 2]] - 1 a[{l_, _}, r_] := ({l + (5r - 3#)/2, #} &)[Mod[r, 2]] a[{l_, 0}, 0] := {l + 1, 0} a[{l_, 1}, 0] := ({(13 + #(5/2)^IntegerExponent[#, 2])/6, 0} &[6l + 2] b[list_, i_] := First[Fold[a, {Apply[Plus, Drop[list, -i]], 0}, Apply[Plus, Split[Take[list, -i], #1  #2 ≠ 0 &], 1]]] (The corresponding expression for t[x] is more complicated.)

It is related to (a) by Gray code reordering of the rows, and to (b) by reordering according to (see page 905 ) BitReverseOrder[a_] := With[{n = Length[a]}, a 〚 Map[FromDigits[Reverse[#], 2] &, IntegerDigits[Range[0, n - 1], 2, Log[2, n]]] + 1 〛 ] It is also given by Array[Apply[Times, (-1)^(IntegerDigits[#1, 2, s] Reverse[IntegerDigits[#2, 2, s]])] &, 2^{s,s}, 0] where (b) is obtained simply by dropping the Reverse .

In the late 1800s it was noted in recreational mathematics that one could find the value of π by looking at randomly dropped needles.

Here are examples of how some of the basic Mathematica constructs used in the notes in this book work: • Iteration Nest[f, x, 3] ⟶ f[f[f[x]]] NestList[f, x, 3] ⟶ {x, f[x], f[f[x]], f[f[f[x]]]} Fold[f, x, {1, 2}] ⟶ f[f[x, 1], 2] FoldList[f, x, {1, 2}] ⟶ {x, f[x, 1], f[f[x, 1], 2]} • Functional operations Function[x, x + k][a] ⟶ a + k (# + k&)[a] ⟶ a + k (r[#1] + s[#2]&)[a, b] ⟶ r[a] + s[b] Map[f, {a, b, c}] ⟶ {f[a], f[b], f[c]} Apply[f, {a, b, c}] ⟶ f[a, b, c] Select[{1, 2, 3, 4, 5}, EvenQ] ⟶ {2, 4} MapIndexed[f, {a, b, c}] ⟶ {f[a, {1}], f[b, {2}], f[c, {3}]} • List manipulation {a, b, c, d} 〚 3 〛 ⟶ c {a, b, c, d} 〚 {2, 4, 3, 2} 〛 ⟶ {b, d, c, b} Take[{a, b, c, d, e}, 2] ⟶ {a, b} Drop[{a, b, c, d, e}, -2] ⟶ {a, b, c} Rest[{a, b, c, d}] ⟶ {b, c, d} ReplacePart[{a, b, c, d}, x, 3] ⟶ {a, b, x, d} Length[{a, b, c}] ⟶ 3 Range[5] ⟶ {1, 2, 3, 4, 5} Table[f[i], {i, 4}] ⟶ {f[1], f[2], f[3], f[4]} Table[f[i, j], {i, 2}, {j, 3}] ⟶ {{f[1, 1], f[1, 2], f[1, 3]}, {f[2, 1], f[2, 2], f[2, 3]}} Array[f, {2, 2}] ⟶ {{f[1, 1], f[1, 2]}, {f[2, 1], f[2, 2]}} Flatten[{{a, b}, {c}, {d, e}}] ⟶ {a, b, c, d, e} Flatten[{{a, {b, c}}, {{d}, e}}, 1] ⟶ {a, {b, c}, {d}, e} Partition[{a, b, c, d}, 2, 1] ⟶ {{a, b}, {b, c}, {c, d}} Split[{a, a, a, b, b, a, a}] ⟶ {{a, a, a}, {b, b}, {a, a}} ListConvolve[{a, b}, {1, 2, 3, 4, 5}] ⟶ {2a + b, 3a + 2b, 4a + 3b, 5a + 4b} Position[{a, b, c, a, a}, a] ⟶ {{1}, {4}, {5}} RotateLeft[{a, b, c, d, e}, 2] ⟶ {c, d, e, a, b} Join[{a, b, c}, {d, b}] ⟶ {a, b, c, d, b} Union[{a, a, c, b, b}] ⟶ {a, b, c} • Transformation rules {a, b, c, d} /. b  p ⟶ {a, p, c, d} {f[a], f[b], f[c]} /. f[a]  p ⟶ {p, f[b], f[c]} {f[a], f[b], f[c]} /. f[x_]  p[x] ⟶ {p[a], p[b], p[c]} {f[1], f[b], f[2]} /. f[x_Integer]  p[x] ⟶ {p[1], f[b], p[2]} {f[1, 2], f[3], f[4, 5]} /. f[x_, y_]  x + y ⟶ {3, f[3], 9} {f[1], g[2], f[2], g[3]} /. f[1] | g[_]  p ⟶ {p, p, f[2], p} • Numerical functions Quotient[207, 10] ⟶ 20 Mod[207, 10] ⟶ 7 Floor[1.45] ⟶ 1 Ceiling[1.45] ⟶ 2 IntegerDigits[13, 2] ⟶ {1, 1, 0, 1} IntegerDigits[13, 2, 6] ⟶ {0, 0, 1, 1, 0, 1} DigitCount[13, 2, 1] ⟶ 3 FromDigits[{1, 1, 0, 1}, 2] ⟶ 13 The Mathematica programs in these notes are formatted in Mathematica StandardForm .