Sierpiński pattern Other ways to generate step n of the pattern shown here in various orientations include: • Mod[Array[Binomial, {2, 2} n , 0], 2] (see pages 611 and 870 ) • 1 - Sign[Array[BitAnd, {2, 2} n , 0]] (see pages 608 and 871 ) • NestList[Mod[RotateLeft[#] + #, 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • NestList[Mod[ListConvolve[{1, 1}, #, -1], 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • IntegerDigits[NestList[BitXor[2#, #] &, 1, 2 n - 1], 2, 2 n ] (see page 906 ) • NestList[Mod[Rest[FoldList[Plus, 0, #]], 2] &, Table[1, {2 n }], 2 n - 1] (see page 1034 ) • Table[PadRight[ Mod[CoefficientList[(1 + x) t - 1 , x], 2], 2 n - 1], {t, 2 n }] (see pages 870 and 951 ) • Reverse[Mod[CoefficientList[Series[1/(1 - (1 + x)y), {x, 0, 2 n - 1}, {y, 0, 2 n - 1}], {x, y}], 2]] (see page 1091 ) • Nest[Apply[Join, MapThread[ Join, {{#, #}, {0 #, #}}, 2]] &, {{1}}, n] (compare page 1073 ) The positions of black squares can be found from: • Nest[Flatten[2# /.

(In Mathematica 4 and above PadLeft[{1}, n, 0, Floor[n/2]] can be used instead.) … RotateLeft[a] and RotateRight[a] make shifted versions of the original list of cell values a . … Note that by using RotateLeft and RotateRight one automatically gets cyclic boundary conditions.

The so-called Paley family of Hadamard matrices for n = 4k = p + 1 with p prime are given by PadLeft[Array[JacobiSymbol[#2 - #1, n - 1]&, {n, n} - 1] - IdentityMatrix[n - 1], {n, n}, 1] Originally introduced by Jacques Hadamard in 1893 as the matrices with elements Abs[a] ≤ 1 which attain the maximal possible determinant ± n n/2 , Hadamard matrices appear in various combinatorial problems, particularly design of exhaustive combinations of experiments and Reed–Muller error-correcting codes.

Implementation [of 2D cellular automata] An n × n array of white squares with a single black square in the middle can be generated by PadLeft[{{1}}, {n, n}, 0, Floor[{n, n}/2]] For the 5-neighbor rules introduced on page 170 each step can be implemented by CAStep[rule_, a_] := Map[rule 〚 10 - # 〛 &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}] where rule is obtained from the code number by IntegerDigits[code, 2, 10] . … In d dimensions with k colors, 5-neighbor rules generalize to (2d + 1) -neighbor rules, with CAStep[{rule_, d_}, a_] := Map[rule 〚 -1 - # 〛 &, a + k AxesTotal[a, d], {d}] AxesTotal[a_, d_] := Apply[Plus, Map[RotateLeft[a, #] + RotateRight[a, #]&, IdentityMatrix[d]]] with rule given by IntegerDigits[code, k, k(2d(k - 1) + 1)] . 9-neighbor rules generalize to 3 d -neighbor rules, with CAStep[{rule_, d_}, a_] := Map[rule 〚 -1 - # 〛 &, a + k FullTotal[a, d], {d}] FullTotal[a_, d_] := Array[RotateLeft[a, {##}] &, Table[3, {d}], -1, Plus] - a with rule given by IntegerDigits[code, k, k((3 d - 1)(k - 1) + 1)] .

Random walks In one dimension, a random walk with t steps of length 1 starting at position 0 can be generated from NestList[(# + (-1)^Random[Integer])&, 0, t] or equivalently FoldList[Plus, 0, Table[(-1)^Random[Integer], {t}]] A generalization to d dimensions is then FoldList[Plus, Table[0, {d}], Table[RotateLeft[PadLeft[ {(-1)^Random[Integer]}, d], Random[Integer, d - 1]], {t}]] A fundamental property of random walks is that after t steps the root mean square displacement from the starting position is proportional to √ t .