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Numbers arranged in a triangle
In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as
Triangular_array
in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrices, or concretely in physical space. It
List_of_triangle_topics
Triangular array of consecutive natural numbers
Floyd's triangle is a triangular array of natural numbers used in computer science education. It is named after Robert Floyd. It is defined by filling
Floyd's_triangle
Triangular array of the binomial coefficients
35\quad 35\quad 21\quad 7\quad 1\end{array}}} In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play
Pascal's_triangle
Type of data structure
(non-linear) formulas are occasionally used. For a compact two-dimensional triangular array, for instance, the addressing formula is a polynomial of degree 2.
Array_(data_structure)
A Riordan array is an infinite lower triangular matrix, D {\displaystyle D} , constructed from two formal power series, d ( t ) {\displaystyle d(t)} of
Riordan_array
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
Numerical integration method
rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand
Romberg's_method
Subclass of mammalia
new clade known as the Australosphenida, and also suggest that the triangular array of cusps may have evolved independently in australosphenidans and therians
Prototheria
Triangular array of natural numbers
\operatorname {N} (n,k),n\in \mathbb {N} ^{+},1\leq k\leq n} form a triangular array of natural numbers, called the Narayana triangle, that occur in various
Narayana_number
Number of subsets of a given size
{\tbinom {n}{n}}} in successive rows for n = 0, 1, 2, ... gives a triangular array called Pascal's triangle, satisfying the recurrence relation ( n k
Binomial_coefficient
Type of probability distribution
a.n.) random variables within a triangular array X 11 X 21 X 22 X 31 X 32 X 33 ⋮ ⋮ ⋮ ⋱ {\displaystyle {\begin{array}{cccc}X_{11}\\X_{21}&X_{22}\\X_{
Infinite divisibility (probability)
Infinite_divisibility_(probability)
Mathematical transformation on sequences
is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to
Boustrophedon_transform
T\right)} -bimodule, then the triangular matrix ring R := [ T 0 M U ] {\displaystyle R:=\left[{\begin{array}{cc}T&0\\M&U\\\end{array}}\right]} consists of 2-by-2
Triangular_matrix_ring
3rd–2nd century BC Indian mathematician and poet
{\displaystyle n} . Mātrā-meru/Meru-prastāra (Pyramidal arrangement): The triangular array of combinatorial binomial coefficients (later known in Europe as Pascal's
Pingala
In combinatorics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers
Rencontres_numbers
Mathematics term
Lozanić's triangle (sometimes called Losanitsch's triangle) is a triangular array related to of Pascal's triangle. It is named after the Serbian chemist
Lozanić's_triangle
Hebrew niqqud vowel sign
Same sound tzere Example שֶׁל The word for of in Hebrew, shel. The triangular array of three dots under the letter Shin form the segol. Other Niqqud Shva ·
Segol
Count of the possible partitions of a set
10 15 15 20 27 37 52 {\displaystyle {\begin{array}{l}1\\1&2\\2&3&5\\5&7&10&15\\15&20&27&37&52\end{array}}} The Bell numbers appear on both the left and
Bell_number
Public library in Kanazawa city, Ishikawa Prefecture, Japan
of glass which puncture the concrete surface of the building in a triangular array. The firm that designed the library, Coelacanth K&H, describe the building
Kanazawa_Umimirai_Library
Number of paths between grid corners, allowing diagonal steps
octahedral numbers. Alternatively, the same numbers can be arranged in a triangular array resembling Pascal's triangle, also called the tribonacci triangle,
Delannoy_number
Type of neuron
fields) which are roughly equal in size and arranged in a periodic triangular array that covers the entire available environment. Cells with this firing
Grid_cell
Radio astronomy observatory in New Mexico, US
The Karl G. Jansky Very Large Array (VLA) is a centimeter-wavelength radio astronomy observatory in the southwestern United States built in the 1970s
Very_Large_Array
Polynomials in combinatorial mathematics
inversion. The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by B n , k ( x 1 , x 2 , … , x n − k + 1 ) = ∑
Bell_polynomials
Persian polymath and poet (1048–1131)
later came to know as Pascal's triangle; Khayyam popularized this triangular array in Iran, so that it is now known as Omar Khayyam's triangle. In 1074–5
Omar_Khayyam
structure that Yablonovitch was able to produce involved drilling a triangular array of cylindrical holes in layers of transparent material, where the holes
Yablonovite
Set of rules defining correctly structured programs
several of which are described below. Take Pascal's triangle, which is a triangular array of numbers in which those at the ends of the rows are 1 and each of
APL_syntax_and_symbols
Mathematical integer sequence
{\displaystyle (0,0)} to ( n , 0 ) {\displaystyle (n,0)} . There is also a triangular array associated with the Schröder numbers that provides a recurrence relation
Schröder_number
Polynomial sequence
n-1} for n > 0 {\textstyle n>0} . A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common
Eulerian_number
Triangular array of integers following a modular formula
of Boston, Massachusetts. In mathematics, the Dudley triangle is a triangular array of integers that was defined by Underwood Dudley (1987). It consists
Dudley_triangle
Data structure used to implement multi-dimensional arrays
used to implement jagged arrays, such as triangular arrays, triangular matrices and other kinds of irregularly shaped arrays. The data structure is named
Iliffe_vector
Numbers parameterizing ways to partition a set
{\displaystyle a_{n}=\sum _{k=0}^{n}k!\left\{{n \atop k}\right\}.} Below is a triangular array of values for the Stirling numbers of the second kind (sequence A048993
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
1997 book about mathematics for children and young adults
Number Devil presents Pascal's triangle and the patterns that the triangular array displays. On the eighth night, Robert is brought to his classroom at
The_Number_Devil
Subclass of mammals
new clade known as the Australosphenida, and also suggest that the triangular array of cusps may have evolved independently in australosphenidans and therians
Yinotheria
Data type that represents an ordered collection of elements (values or variables)
can be used for triangular matrices and other oddly-shaped arrays. In order to effectively implement variables of such types as array structures (with
Array_(data_type)
Type of surface-mount packaging for integrated circuits
The land grid array (LGA) is a type of surface-mount packaging for integrated circuits (ICs) that is notable for having the pins on the socket (when a
Land_grid_array
Musical artist
Partch' tonality diamond is related to the harmonic series when put in a triangular array. Landman co-wrote Nice Noise (about prepared guitar techniques and
Yuri_Landman
Clear-channel AM radio station in New York City
location features three full half-wave (692 feet) guyed antennas in a triangular array. WOR was within one mile of both AM 1190 WLIB and AM 1010 WINS. Thus
WOR_(AM)
Algorithm for solving systems of linear equations
Gaussian elimination refers to the process until it has reached its upper triangular, (unreduced) row echelon form. For computational reasons, when solving
Gaussian_elimination
Chemical compound
variety of substrates. The trinuclear cluster features an isosceles triangular array of metals with one short edge (rOs-Os = 2.68 Å), which is spanned by
Decacarbonyldihydridotriosmium
Decacarbonyldihydridotriosmium
Sum of a factorial number and a triangular number
generalization gives rise to factoriangular triangles, which are Pascal-like triangular arrays of numbers. Two such triangles can be formed: A triangle with entries
Factoriangular_number
Cargo ship of the United States Navy (1957–1990)
implementation of a short baseline acoustic system, is composed of a triangular array of hydrophones on the ship's hull, a transponder on the towed vehicle
USNS_Mizar
Overview of and topical guide to discrete mathematics
Empty product – Result from multiplying no factors Pascal's triangle – Triangular array of the binomial coefficients Combinatorial proof – Proofs in enumerative
Outline of discrete mathematics
Outline_of_discrete_mathematics
According to Reck et al., the practical implementation of this scheme is a triangular array of beam splitters and phase-shifters. This is where each beam splitter
Universal multiport interferometer
Universal_multiport_interferometer
{\displaystyle \textstyle h} -vector of Δ {\displaystyle \Delta } , construct a triangular array by first writing d + 2 {\displaystyle d+2} 1 {\displaystyle \textstyle
H-vector
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns
Pentagonal_number
Chemical compound composed of boron and hydrogen atoms only
revealed the prevalence of deltahedral structures, i.e., networks of triangular arrays of BH centers. The bonding of the clusters ushered in Polyhedral skeletal
Boron_hydride_clusters
Historic Mamluk-era monument in Cairo, Egypt
through an array of muqarnas squinches. The muqarnas have three tiers of cells and are interrupted in the middle of each wall by a triangular array of windows
Tomb of Salar and Sangar al-Gawli
Tomb_of_Salar_and_Sangar_al-Gawli
Type of matrix factorization
factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition)
LU_decomposition
Ordered array of colloidal particles
themselves into highly ordered arrays. Rod-shaped particles in the tobacco mosaic virus could form a two-dimensional triangular lattice, while a body-centered
Colloidal_crystal
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and
Matrix_(mathematics)
Property of geometry, also used to generalize the notion of "distance" in metric spaces
degenerate. The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally
Triangle_inequality
Polygon mesh composed of triangles
index arrays, a mesh is represented by two separate arrays, one array holding the vertices, and another holding sets of three indices into that array which
Triangle_mesh
Global bioclimatic scheme for the classification of land areas
scheme. Although conceived as three-dimensional by its originator, it is usually shown as a two-dimensional array of hexagons in a triangular frame.
Holdridge_life_zones
Two-dimensional cellular automaton
Golly are used. These programs often do not store cells as two-dimensional arrays, instead using algorithms such as Hashlife which represent patterns as a
Conway's_Game_of_Life
Solid made by joining an n- and 2n-gon with triangles and squares
squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and
Cupola_(geometry)
Natural number
2016 = 25 × 32 × 7. It is the second-smallest Erdős–Nicolas numberand a triangular number. 2017 is a sexy prime with 2011. 2024 = 23 × 11 × 23. It is a tetrahedral
2000_(number)
Group of protists
with radially directed axopods, supported by microtubules in a triangular-hexagonal array arising from an amorphous central granule. There are only three
Gymnosphaerid
Concept developed by C. Hart Merriam in 1889
classification scheme. Although conceived as three-dimensional by its originator, is usually shown as a two-dimensional array of hexagons in a triangular frame.
Life_zone
Former radio telescope in Puerto Rico, United States
Kilometre Array (SKA, Australia, South Africa) Submillimeter Array (SMA, US) Very Large Array (VLA, New Mexico, US) Very Long Baseline Array (VLBA, US)
Arecibo_Telescope
Algorithm for generating contour lines on a 2D scalar field
that generates contours for a two-dimensional scalar field (rectangular array of individual numerical values). A similar method can be used to contour
Marching_squares
Possible form of a matrix
upper triangular matrix with nonzero entries on the diagonal is in row echelon form, and a matrix in row echelon form is (weakly) upper triangular. Thus
Row_echelon_form
Function used in signal processing
rectangular window. They include the rectangular window itself (k = 1), the § Triangular window (k = 2) and the § Parzen window (k = 4). Alternative definitions
Window_function
center bearing three red six-pointed stars outlined in green arranged in a triangular design (one star above, two stars below). Cambodia Cambodia 2:3 June 30
List of national flags of sovereign states
List_of_national_flags_of_sovereign_states
mirror that map all three end nodes together. The vertex figure is a triangular prism, containing two cubes and three cuboctahedra. The three symmetries
Rectified_24-cell
Numbers obtained by adding the two previous ones
that 8 and 144 are the only such non-trivial perfect powers. The only triangular Fibonacci numbers are 1, 3, 21, and 55, which was conjectured by Vern
Fibonacci_sequence
a numeric solution method. 0 0 1 {\displaystyle {\begin{array}{c|c}0&0\\\hline &1\\\end{array}}} Second-order methods can be generically written as follows:
List_of_Runge–Kutta_methods
Family of implicit and explicit iterative methods
&a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\\end{array}}={\begin{array}{c|c}\mathbf {c} &A\\\hline &\mathbf {b^{T}} \\\end{array}}} The consequence of this difference
Runge–Kutta_methods
Radio telescope in Guizhou, China
of steel cables hanging from the rim. FAST's surface is made of 4,450 triangular panels, 11 m (36 ft) on a side, in the form of a geodesic dome. There
Five-hundred-meter Aperture Spherical Telescope
Five-hundred-meter_Aperture_Spherical_Telescope
Array that replaces runtime computation with a simpler array indexing operation
science, a lookup table (LUT) is an array that replaces runtime computation of a mathematical function with a simpler array indexing operation, in a process
Lookup_table
Rare genetic condition
lower jaw bone (mandible) is positioned further back than normal Ears Triangular ears Low positioned, upward pointing ears Eyes Hypertelorism - widely
Nablus mask-like facial syndrome
Nablus_mask-like_facial_syndrome
Fuzzy logic concept
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces
T-norm
Biogeographical unit with a particular biological community
scheme. Although conceived as three-dimensional by its originator, it is usually shown as a two-dimensional array of hexagons in a triangular frame.
Biome
Natural number
to 38, or twice 19. A hexaflexagon is a strip of nineteen alternating triangular faces that can flex into a regular hexagon, such that any two of six colorings
19_(number)
Coordinate system that is defined by points instead of vectors
1 : 0 C = 0 : 0 : 1 {\displaystyle {\begin{array}{rccccc}A=&1&:&0&:&0\\B=&0&:&1&:&0\\C=&0&:&0&:&1\end{array}}} The centroid would be at 1 3 : 1 3 : 1 3
Barycentric_coordinate_system
Array of partial sums of the binomial coefficients
Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0
Bernoulli's_triangle
Effort to find civilizations not from Earth
an Earth-based radio telescope array with 1,500 dishes known as "Project Cyclops". The price tag for the Cyclops array was US$10 billion. Cyclops was
Search for extraterrestrial intelligence
Search_for_extraterrestrial_intelligence
Whole number
of an array are numbered starting from 0 in C, so that for an array of n items the sequence of array indices runs from 0 to n−1. Since C arrays are accessed
0
Type of roof where all sides slope downward to the walls
meeting at a central ridge), Pyramid Hip Roof (a square base where all four triangular sides meet at a single peak), Cross Hip Roof (with multiple intersecting
Hip_roof
Polyhedron with 62 faces
its edge arrangement with the great dodecicosidodecahedron (having the triangular and pentagrammic faces in common), and the great rhombidodecahedron (having
Nonconvex great rhombicosidodecahedron
Nonconvex_great_rhombicosidodecahedron
Matrix defined using smaller matrices called blocks
{\displaystyle \left[{\begin{array}{ccc|c}a_{11}&a_{12}&a_{13}&b_{1}\\a_{21}&a_{22}&a_{23}&b_{2}\\\hline c_{1}&c_{2}&c_{3}&d\end{array}}\right]} Any matrix may
Block_matrix
Features that define a (ship) sail's shape and function
to the vessel (e.g. fore-and-aft) and its shape, (e.g. (a)symmetrical, triangular, quadrilateral, etc.). Sails are typically constructed out of flexible
Sail_components
Number equal to the sum of its proper divisors
even perfect number is the ( 2 p − 1 ) {\displaystyle (2^{p}-1)} -th triangular number (and hence equal to the sum of the integers from 1 to 2 p − 1
Perfect_number
Semiconductor light source
frequency, they can create a phantom array, where ghost images of the LED appear if the eyes quickly scan across the array. White LED headlamps are beginning
Light-emitting_diode
Visualization of the prime numbers
discovery, the herpetologist Laurence Klauber constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration
Ulam_spiral
Complex number whose mapping on a coordinate plane produces a triangular lattice
primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers,
Eisenstein_integer
Airborne system of surveillance radar plus command and control functions
used an active electronically scanned array (AESA) – an active phased array radar. This radar consists of an array of transmit/receive (T/R) modules that
Airborne early warning and control
Airborne_early_warning_and_control
Approximation of a matrix's Cholesky factorization
N is A = LL* where L is a lower triangular matrix. An incomplete Cholesky factorization is given by a lower triangular matrix K that is sparser than L
Incomplete Cholesky factorization
Incomplete_Cholesky_factorization
shape consists of 3 sides and is one of the simplest types of mesh. A triangular surface mesh is always quick and easy to create. It is most common in
Types_of_mesh
Signal processing method
proposed for frequency estimation. However, with the introduction of phased-array systems in everyday technology, it is also used for angle of arrival estimations
Estimation of signal parameters via rotational invariance techniques
Estimation_of_signal_parameters_via_rotational_invariance_techniques
Associative algebra used in combinatorics
usual order on the indices of the matrices, they will appear as upper-triangular matrices with a prescribed zero-pattern determined by the incomparable
Incidence_algebra
Intersection of triangle altitudes
: 0 {\displaystyle {\begin{array}{rccccc}D=&0&:&\sec B&:&\sec C\\E=&\sec A&:&0&:&\sec C\\F=&\sec A&:&\sec B&:&0\end{array}}} The extended sides of the
Orthocenter
Iterative method used to solve a linear system of equations
{A} } is decomposed into a lower triangular component L {\displaystyle \mathbf {L} } , and a strictly upper triangular component U {\displaystyle \mathbf
Gauss–Seidel_method
Elementwise product of two matrices
\quad {B}=\left[{\begin{array}{c | c | c }{B}_{1}&{B}_{2}&{B}_{3}\end{array}}\right]=\left[{\begin{array}{c c c | c c c | c c c
Hadamard_product_(matrices)
Matrix equal to its transpose
positive-definite symmetric matrix A {\displaystyle A} is a product of a lower-triangular matrix L {\displaystyle L} and its transpose, A = L L T . {\displaystyle
Symmetric_matrix
Species of flowering plant
or roughly triangular blades up to 15 centimeters (6 in) long which have smooth edges or faint lobes or teeth. The inflorescence is an array of several
Proboscidea_parviflora
Neighborhood of Atlanta in Fulton, Georgia, United States
route of the city of Atlanta’s scenic Christmas array of lights. The neighborhood has two triangular Parks, I and II, with Park I dedicated to its first
Just_Us,_Atlanta
4-D polyhedral pyramid
and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one, the triangular pyramids
Octahedral_pyramid
Graph in which every two vertices are adjacent
contributions of Kazimierz Kuratowski to graph theory. Kn has n(n − 1)/2 edges (a triangular number), and is a regular graph of degree n − 1. All complete graphs are
Complete_graph
Species of flowering plant
oval, or roughly triangular blades up to 7 centimeters long which are deeply lobed and wavy along the edges. The inflorescence is an array of many showy
Proboscidea_althaeifolia
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