(One can either determine frequencies of notes directly from the values of elements, or, say, from cumulative sums of such values, or from heights in paths like those on page 892 .)

An example suggested by Stanislaw Ulam around 1960 (in a peculiar attempt to get a 1D analog of a 2D cellular automaton; see pages 877 and 928 ) starts with {1, 2} , then successively appends the smallest number that is the sum of two previous numbers in just one way, yielding {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, …} With this initial condition, the sequence is known to go on forever.

Note that isotropy can also be characterized using analogs of multipole moments, obtained in 2D by summing r i Exp[  n θ i ] , and in higher dimensions by summing appropriate SphericalHarmonicY or GegenbauerC functions. … (Sums of squares of moments of given order in general provide rotationally invariant measures of anisotropy—equal to pair correlations weighted with LegendreP or GegenbauerC functions.)

Nested behavior is also found for example in EllipticTheta[3, 0, z] , which is given essentially by Sum[z n 2 , {n, ∞ }] .

Statements in Peano arithmetic Examples include: • √ 2 is irrational: ¬ ∃ a ( ∃ b (b ≠ 0 ∧ a × a  ( Δ Δ 0) × (b × b))) • There are infinitely many primes of the form n 2 + 1 : ¬ ∃ n ( ∀ c ( ∃ a ( ∃ b (n + c) × (n + c) + Δ 0  ( Δ Δ a) × ( Δ Δ b)))) • Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135 ): ∀ a ( ∃ b ( ∃ c (( Δ Δ 0) × ( Δ Δ a)  b + c ∧ ∀ d ( ∀ e ( ∀ f ((f  ( Δ Δ d) × ( Δ Δ e) ∨ f  Δ 0) ⇒ (f ≠ b ∧ f ≠ c))))))) The last two statements have never been proved true or false, and remain unsolved problems of number theory.

The following definition also handles the more general case of r neighbors: CAStep[TotalisticCARule[rule_List, r_Integer], a_List] := rule 〚 -1 - Sum[RotateLeft[a, i], {i, -r, r}] 〛 One can generate the representation of totalistic rules used by these functions from code numbers using ToTotalisticCARule[num_Integer, k_Integer, r_Integer] := TotalisticCARule[IntegerDigits[num, k, 1 + (k - 1)(2r + 1)], r]

Things appear somewhat simpler with boiling points, and as noticed by Harry Wiener in 1947 (and increasingly discussed since the 1970s) these tend to be well fit as being linearly proportional to the so-called topological index given by the sum of the smallest numbers of connections visited in getting between all pairs of carbon atoms in an alkane molecule.

Nested structure of attractors Associating with each sequence of length n (and k possible colors for each element) a number Sum[a[i] k -i , {i, n}] , the set of sequences that occur in the limit n  ∞ forms a Cantor set.

One can also consider numbers obtained from infinite sums (or by solving recurrence equations). If f[n] is a rational function, Sum[f[n], {n, ∞ }] must just be a linear combination of PolyGamma functions, but again the multivariate case can be much more complicated.

The number of nodes at distance up to r from a given node is at most 1 + Sum[c[i] + c[i - 1], {i, n}] where c[i_] := 2^DigitCount[i, 2] .