A word formed recursively by concatenating the previous two words in a specified order, for example $W_n=W_{n-2}{W}_{n-1}$ from two seed words. Lengths satisfy the Fibonacci recurrence. To locate a symbol, compare its index with the first block's length, choose the containing block, and subtract the first length when entering the second block. This avoids constructing an exponentially long word.

The sequence $\lfloor n\alpha \rfloor$ for $n=1,2,\dots$, with a fixed positive irrational number $\alpha$. Complementary Beatty sequences for $\alpha ,\beta >1$ partition the positive integers when $1/\alpha +1/\beta =1$.

For a nonnegative integer $n$, let $r(n)$ count overlapping occurrences of $11$ in its binary expansion. The Rudin-Shapiro sign sequence is $b(n)=(-1{)}^{r(n)}$. Appending a binary digit gives $b(2n)=b(n)$ and $b(2n+1)=(-1{)}^{n}b(n)$.

Each term is obtained by reading the lengths and values of consecutive runs in the previous term. Starting from $1$ gives $1,11,21,1211,111221,\dots$: for example, $21$ contains one $2$ followed by one $1$, so its successor is $1211$.

A longest common subsequence of two finite sequences has maximum length among sequences that occur in both with their original order. Characters can be skipped; this differs from requiring a contiguous substring. For $ABCBDAB$ and $BDCABA$, $BCBA$ is a common subsequence of length four.

Let $L(i,j)$ be the optimum for the first $i$ and $j$ terms. Set $L(0,j)=L(i,0)=0$. If the final terms agree, $L(i,j)=1+L(i-1,j-1)$; otherwise $L(i,j)=\max (L(i-1,j),L(i,j-1))$. Filling the table gives $O(mn)$ time. Retaining predecessor choices reconstructs an actual subsequence, while two rows suffice if only the length is needed.

A shortest common supersequence of two finite sequences is a shortest sequence containing each as a subsequence. Order is retained, but occurrences need not be contiguous. For $ab$ and $ac$, $abc$ is such a sequence; $acb$ is another.

For lengths $m,n$, its length is $m+n-L$, where $L$ is the longest common subsequence length. Align a common subsequence of maximal length and merge the unmatched portions to prove the upper bound. Conversely, an alignment inside any common supersequence identifies at least $m+n-\ell$ shared positions when its length is $\ell$, proving the lower bound. This formula is for two inputs; several inputs require a different analysis.

A one-dimensional Kronecker sequence is the sequence of fractional parts $\{n\alpha \}$ for $n\ge 0$. If $\alpha$ is rational, it is periodic. If $\alpha$ is irrational, its terms are equidistributed sequence: the proportion lying in $[a,b)$ tends to $b-a$.

For any nonzero integer $k$, the average $N^{-1}{\sum }_{n=0}^{N-1}{e}^{2\pi ikn\alpha }$ tends to zero by the finite geometric series, since $e^{2\pi ik\alpha }\ne 1$. The Weyl criterion converts these cancellations into uniform distribution. In several dimensions, use $\{n\alpha \}$ coordinatewise; uniform distribution requires that $1,{\alpha }_1,\dots ,{\alpha }_d$ be linearly independent over the rational numbers.

A sequence is eventually periodic if there are integers $N\ge 0$ and $T>0$ such that $u_{n+T}=u_n$ for all $n\ge N$. It consists of a finite initial segment followed by a repeating period. Checking finitely many repeated values can suggest eventual periodicity, but a proof requires a reason the repetition continues, such as recurrence of a complete state in a deterministic finite-state process.

A longest nondecreasing subsequence selects indices $i_1<\cdots <i_k$ with $a_{i_1}\le \cdots \le a_{i_k}$ and makes $k$ as large as possible. Selected terms need not be contiguous. In $3,1,2,2,0$, the terms $1,2,2$ form one of length three.

An $O(n\log n)$ algorithm stores the smallest possible final value for each attainable length. For each new value $x$, binary-search for the first stored tail strictly greater than $x$ and replace it, or append if none exists. Equal values therefore extend the sequence. For the strictly increasing variant, search for the first tail greater than or equal to $x$ instead; changing this comparison changes the problem.

A sequence equal to its own sequence of consecutive run lengths. The classical sequence uses the alphabet $\{1,2\}$ and begins $1,2,2,1,1,2,1,2,2,\dots$; variants use other pairs of positive integers.

A ray's intercept time subtracts its tangent travel-time contribution at the receiver range. For a smooth branch, $dT/dx=p$ implies $d\tau /dp=-x$. Thus an intercept-time curve encodes range as its negative slope. Separate turning branches around a low-velocity zone must be retained rather than joined by fictitious rays.

A member of a seed-plant lineage whose ovules are not enclosed in angiosperm carpels. Conifers contribute extensively to forest carbon pools, but the lineage includes different growth forms and reproductive arrangements.

The number of children of a node in a search tree, or an upper bound on that number over the tree. With branching factor at most $b$ and depth at most $d$, a complete traversal visits at most $1+b+\cdots +b^d$ nodes.

For an incoming timelike Schwarzschild geodesic with asymptotic Lorentz factor $k\ge 1$, the critical unstable circular orbit obeysAt $k=1$, $r_c=4\mu$ and $|h_c|=4\mu c$. For $k>1$, the critical radius lies between $3\mu$ and $4\mu$ and $|h_c|>4\mu c$. The marginally bound angular-momentum bound remains sufficient at larger energies but is no longer the exact threshold. Origin at infinity and an inward initial branch are essential; an orbit already inside a barrier is a different initial value problem.

For regular low-energy elastic scattering by a finite-range central potential, higher partial waves are suppressed by their centrifugal barriers. Matching the regular interior radial solution to small-argument exterior spherical Bessel functions gives the displayed scaling of scattering phase shift, in the generic absence of a threshold singularity. Since the channel amplitude is proportional to $\sin {\delta }_{\ell }/k$, it scales as $k^{2\ell }$. The $\ell =0$ term is isotropic and normally leads. A tuned vanishing leading amplitude or a threshold resonance requires examining the next terms rather than applying the generic statement blindly.

The zero-energy parameter of the angular-momentum-zero partial wave. For a short-range interaction in three dimensions, its leading elastic scattering amplitude is $f_0(k)\simeq -a_s/(1+ik{a}_s)$ when effective-range corrections are negligible. For identical dilute bosons this parameter fixes the leading contact coupling $g_B=4\pi {\hslash }^2{a}_s/m_B$.

This zero-range boundary condition replaces a short-range interaction by its S-wave scattering length. If $\psi =A/r+B+O(r)$, it requires $B=-A/a$. An outgoing spherical wave also has a constant $ik$ term; retaining that term gives the unitary scattering-length amplitude $f_0=-a/(1+ika)$. The model neglects effective-range corrections and is applied outside the microscopic range, not literally inside a finite potential.

Scanning transmission electron microscopy forms an image by scanning a focused electron probe across a thin specimen and recording transmitted or scattered electrons at each probe position. The detector's angular range determines which interaction contributes to the recorded brightness.

A detector including the direct beam gives a different image contrast mechanism from an annular detector collecting scattered electrons. High-angle annular dark-field imaging can emphasize composition and mass-thickness differences, but specimen thickness and channeling still affect interpretation. The probe size limits spatial detail, whereas the displayed-to-scanned length ratio sets magnification; increasing the latter cannot sharpen the probe.

The fraction of incident electrons emerging from a specimen as backscattered electrons.

An Everhart–Thornley detector converts emitted electrons into an amplified optical signal in a scanning electron microscope. A positively biased collector attracts low-energy secondary electrons toward a scintillator; their impact produces light carried through a light guide to a photomultiplier.

The measured brightness depends on how many emitted electrons reach the collector, not just on how many leave the surface. Changing specimen tilt or collector bias can therefore change an image without changing composition. Comparing such settings helps identify collection-geometry effects in secondary-electron topographic contrast. ^[ref]