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In recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non-primitive recursive functions
Double_recursion
Quickly growing function
function. Computability theory Double recursion Fast-growing hierarchy Goodstein function Primitive recursive function Recursion (computer science) with parameter
Ackermann_function
American computer scientist
on compiling the language. He invented the continuation to solve a double recursion problem for one of the users of his Lisp implementation. In 1962, Russell
Steve Russell (computer scientist)
Steve_Russell_(computer_scientist)
Function computable with bounded loops
recursive functions. Grzegorczyk hierarchy Recursion (computer science) Primitive recursive functional Double recursion Primitive recursive set function Primitive
Primitive_recursive_function
Theory of computer sciences
In the formal language theory of computer science, left recursion is a special case of recursion where a string is recognized as part of a language by the
Left_recursion
Type of software bug
allowing infinite recursion of a specific sort—tail recursion—to occur without stack overflow. This works because tail-recursion calls do not take up
Stack_overflow
2 {\displaystyle \Sigma ^{2}} having a double pole on the diagonal and no residue. The topological recursion is then a recursive definition of infinite
Topological_recursion
Type of grammar for describing formal languages
parsing, left recursion must be eliminated. Direct recursion, be that left or right, is important in context-free grammars, because there recursion is the only
Parsing_expression_grammar
Recursion without calling a function by name
In computer science, anonymous recursion is recursion which does not explicitly call a function by name. This can be done either explicitly, by using a
Anonymous_recursion
Muran language
researchers, numerals and counting are based on recursion in the language, the absence of recursion in their language entails a lack of counting. That
Pirahã_language
Size of a mathematical ball
{d}{dR}}V_{n}(R)={\frac {n}{R}}V_{n}(R).} Thus, An − 1(R) inherits formulas and recursion relationships from Vn(R), such as A n − 1 ( R ) = 2 π n / 2 Γ ( n 2 )
Volume_of_an_n-ball
Subfield of linguistic semantics
Structure Hypothesis and Verb Phrase Hypothesis, both outlined below. The recursion found under the "umbrella" Verb Phrase, the VP Shell, accommodated binary-branching
Lexical_semantics
Sequence of program instructions invokable by other software
suspend while another nested execution of the same callable executes. Recursion is a useful means to simplify some complex algorithms and break down complex
Function (computer programming)
Function_(computer_programming)
Algorithmic technique
(roughly) every N/2 inputs if the recursion stops at exactly n = N. By making N sufficiently large, the overhead of recursion can be made negligible (precisely
Pairwise_summation
Study of circular causal processes
transdisciplinary study of circular causal processes such as feedback and recursion, where the outcomes of actions return as inputs for subsequent actions
Cybernetics
Integers occurring in the coefficients of the Taylor series of 1/cosh t
denotes the rising factorial. The Euler numbers can be defined by the recursion E 2 n = − ∑ k = 1 n ( 2 n 2 k ) E 2 ( n − k ) , {\displaystyle E_{2n}=-\sum
Euler_numbers
Family of higher-order functions
second argument, and the rest of the result is never demanded, then the recursion will stop (e.g., head == foldr (\a b->a) (error "empty list")). This allows
Fold_(higher-order_function)
Web browser developed by Google
performed poorly on recursion-intensive benchmarks, such as those of Google, because the Mozilla team had not implemented recursion-tracing yet. Two weeks
Google_Chrome
2022 video game
5 on 26 July 2023. The game centers around playing with infinity and recursion by pushing boxes into and out of other boxes, sometimes in and out of
Patrick's_Parabox
Type of matrix factorization
is −1. Matrix formulae to calculate rows and columns of LU factors by recursion are given in the remaining part of Banachiewicz's paper as Eq. (2.3) and
LU_decomposition
Algorithm in computer graphics to add color or texture
stack space is severely constrained (e.g. Microcontrollers). Moving the recursion into a data structure (either a stack or a queue) prevents a stack overflow
Flood_fill
Recursively defined sequence of continuous closed plane fractal curves
an even number of times and you halve the length of the line at each recursion. If you recur to an odd depth (order is odd) then you end up turned 60
Sierpiński_curve
Hybrid sorting algorithm
performance. It begins with quicksort, it switches to heapsort when the recursion depth exceeds a level based on (the logarithm of) the number of elements
Introsort
Infinitely detailed mathematical structure
efficiently than an ordinary line. Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment
Fractal
Product of numbers from 1 to n
by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion adds in a geometric series to O ( n log
Factorial
Overview of and topical guide to logic
function Recursion (computer science) Recursive language Recursive set Recursively enumerable language Recursively enumerable set Reduction (recursion theory)
Outline_of_logic
Polynomial sequence
{He} _{n}'(x).} Individual coefficients are related by the following recursion formula: a n + 1 , k = { − ( k + 1 ) a n , k + 1 k = 0 , a n , k − 1 −
Hermite_polynomials
Type of Knowledge Graph Management System
knowledge graphs also makes the presence of recursion in these rules a particularly important aspect. Recursion means that the same rules might be called
Vadalog
Fractal named after mathematician Benoit Mandelbrot
connectedness locus has also been studied; here one considers the two-parameter recursion z ↦ z 3 + 3 k z + c {\displaystyle z\mapsto z^{3}+3kz+c} , whose two critical
Mandelbrot_set
General-purpose programming language
language, supporting structured programming, lexical variable scope, and recursion, with a static type system. It was designed to be compiled to provide
C_(programming_language)
Sphere with radius one, usually centered on the origin of the space
displayed precision. The A n {\displaystyle A_{n}} values satisfy the recursion: A 0 = 2 {\displaystyle A_{0}=2} A 1 = 2 π {\displaystyle A_{1}=2\pi }
Unit_sphere
General-purpose programming language
FORTRAN II added support for the DOUBLE PRECISION and COMPLEX data types. Early FORTRAN compilers supported no recursion in subroutines. Early computer
Fortran
1979 book by Douglas Hofstadter
another, but slower and negated. The book contains many instances of recursion and self-reference, where objects and ideas speak about or refer back
Gödel,_Escher,_Bach
Element mapped to itself by a mathematical function
extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory
Fixed_point_(mathematics)
Functional programming language
-> a -- Using recursion (with the "ifthenelse" expression) factorial n = if n < 2 then 1 else n * factorial (n - 1) -- Using recursion (with pattern matching)
Haskell
Method to evaluate polynomials in Bernstein form
_{1}\\[1ex]\beta _{2}^{(0)}&=\beta _{2}\end{aligned}}} at the point t0. We start the recursion with β 0 ( 1 ) = β 0 ( 0 ) ( 1 − t 0 ) + β 1 ( 0 ) t 0 = β 0 ( 1 − t 0
De_Casteljau's_algorithm
Programming style in which control is passed explicitly
will cause both the constructed continuation to potentially grow during recursion, and the call stack. This is usually undesirable, but has been used in
Continuation-passing_style
Browser-based graphing calculator
"High-iteration series and infinite integration examples". Desmos. "Recursion – Desmos Help Center". help.desmos.com. April 22, 2024. Retrieved May
Desmos
Axiomatic set theories based on the principles of mathematical constructivism
{\displaystyle g(Sn)=f(g(n))} . This iteration- or recursion principle is akin to the transfinite recursion theorem, except it is restricted to set functions
Constructive_set_theory
Logical operation
used precedence of logical operators. Within a system of classical logic, double negation, that is, the negation of the negation of a proposition P {\displaystyle
Negation
Discrete Fourier transform algorithm
traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley–Tukey algorithm breaks the DFT into smaller
Fast_Fourier_transform
Coefficients coupled with angular momentum
group SO(3). When "spin" angular momenta are included, the group is its double covering group, SU(2). A reducible representation is one where a change
3-j_symbol
Programming language that uses first order logic
called tail call optimization for deterministic predicates exhibiting tail recursion or, more generally, tail calls: A clause's stack frame is discarded before
Prolog
Fast Fourier Transform algorithm
implementations the depth-first recursion is eliminated in favor of a nonrecursive breadth-first approach, although depth-first recursion has been argued to have
Cooley–Tukey_FFT_algorithm
2010 studio album by Tame Impala
original image has been digitally altered using the Droste effect in recursion to make it appear as if the image continues into itself, creating a distinctly
Innerspeaker
Object which stores memory addresses in a computer program
support for pointers to all data types (including pointers to structures), recursion, multitasking, string handling, and extensive built-in functions. PL/I
Pointer (computer programming)
Pointer_(computer_programming)
Sentence, idea or formula that refers to itself
varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation) in functional programming
Self-reference
Fractal composed of triangles
2^{d}} congruent copies that fit inside the enlarged figure: for instance, doubling a line segment (1-dimensional) yields 2 copies, a square (2-dimensional)
Sierpiński_triangle
2010 film by Christopher Nolan
and the splinter suffix -ception, which refer to layering, nesting, or recursion, in reference to the movie's key element of a "dream within a dream".
Inception
Process of embedding one phrase in another
are located inside the sentence spaced throughout. Garden path sentence Recursion Complementizer KARLSSON, FRED (2007-06-18). "Constraints on multiple center-embedding
Center_embedding
One That I Want" from Grease.[citation needed] "recursion( see it )" shows a "Did you mean: recursion", linking to the same search over and over again
List_of_Google_Easter_eggs
General-purpose programming language
commonly provide tail call optimization to allow for extensive use of recursion without stack overflow problems. Limitations in JVM bytecode complicate
Scala_(programming_language)
by the phrase "braid groups are linear." Recursion theory Rózsa Péter was one of the founders of recursion theory, a branch of mathematical logic, of
List of inventions and discoveries by women
List_of_inventions_and_discoveries_by_women
Abstract machine used in a formal logic and theoretical computer science
function) Successor function Identity function Composition function Primitive recursion (induction) μ operator (unbounded search operator) The authors show that
Counter_machine
Programming language: compiled, structured, array language
require a result clause that returns the value of a function. F supports recursion. All of the intrinsic procedures found in Fortran 95 may be used in F
F_(programming_language)
Programming language
features a static type system, type inference, parametric polymorphism, tail recursion, pattern matching, first class lexical closures, functors (parametric
OCaml
Programming language used in Texas Instruments calculators
pausing execution and they cannot store specific results into variables. Recursion is possible. A program can be called from within itself or from within
TI-BASIC
Application of cybernetics to itself
nature of human knowledge Double hermeneutic – a feature of social sciences, that they can influence the subject of their study Double-loop learning – the modification
Second-order_cybernetics
Topics referred to by the same term
Degree of a character in representation theory Degree of unsolvability in recursion theory Degree of a central simple algebra Degree of a permutation group
Degree
Israeli military operation against Syria
men in Daraa". SOHR. 4 November 2025. Retrieved 7 January 2026. "New recursion: Israeli forces infiltrate Khan Arnaba and arrest four members of a family"
Israeli invasion of Syria (2024–present)
Israeli_invasion_of_Syria_(2024–present)
Functions of an angle
previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. Being defined as fractions
Trigonometric_functions
Numbers in a type of Lucas sequence
}}n>1.\\\end{cases}}} The next Jacobsthal number is also given by the recursion formula J n + 1 = 2 J n + ( − 1 ) n , {\displaystyle J_{n+1}=2J_{n}+(-1)^{n}
Jacobsthal_number
spreadsheet are usually called "rows". recursion Occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines
Glossary_of_computer_science
operators, user defined constants, user defined functions, user defined recursion, Unicode mathematical symbols support. Source: mXparser supports basic
MXparser
Method of deriving conclusions
Rules of replacement are sometimes expressed using a double semi-colon. For instance, the double negation rule can be written as P :: ¬ ¬ P {\displaystyle
Rule_of_inference
Binary tree variant
allowing tree traversal without recursion and the extra storage (proportional to the tree's depth) that recursion requires. "A binary tree is threaded
Threaded_binary_tree
Numerical integration method
*/ double romberg(double (*f)(double), double a, double b, size_t max_steps, double acc) { double R1[max_steps], R2[max_steps]; // buffers double *Rp
Romberg's_method
primitive recursion A form of recursion where a function is defined in terms of itself, using simpler cases, with a base case to stop the recursion. primitive
Glossary_of_logic
Social deduction game
accidentally by the werewolves, the seer, or the moderator Derivation is by recursion on decreasing numbers of players in the following round, see: Yao, E.
Mafia_(party_game)
Concept in theoretical computer science
for electrical engineers and technical specialists. Discusses recursion, partial-recursion with reference to Turing Machines, halting problem. A reference
Busy_beaver
General-purpose programming language
documentation. Retrieved 30 April 2026. van Rossum, Guido (22 April 2009). "Tail Recursion Elimination". Neopythonic.blogspot.be. Archived from the original on 19
Python_(programming_language)
American television series
ninth season of the animated series Archer, "Danger Island". A fictional recursion occurs in "The Sultan of Swat" in which – while waiting for the Boeing
Tales_of_the_Gold_Monkey
Foundational controversy in twentieth-century mathematics
This is in fact the so-called "induction schema" used in the notion of "recursion" that was still in development at this time (van Heijenoort p. 493). This
Brouwer–Hilbert_controversy
Programming language
that combine with primitive functions to perform types of iteration and recursion. Therefore, complex and extended transformations of a dataset can be expressed
K_(programming_language)
Text processor used with C and C++ and other programming tools
trying to include the standard library <stdio.h> would cause an infinite recursion of including if using #include, as it would re-include itself. #include_next
C_preprocessor
Problem optimization method
sub-problems. Such optimal substructures are usually described by means of recursion. For example, given a graph G=(V,E), the shortest path p from a vertex
Dynamic_programming
Mathematical functions
multiplication formulas for M {\displaystyle M} and N {\displaystyle N} by recursion. Gauss' M {\displaystyle M} and N {\displaystyle N} satisfy the following
Lemniscate_elliptic_functions
Numbers obtained by adding the two previous ones
steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization). Most identities involving Fibonacci numbers can be
Fibonacci_sequence
Humor that alludes to itself
in English derived from Greek Meta-reference – Type of self reference Recursion – Process of repeating items in a self-similar way Self-reference – Sentence
Self-referential_humor
Algorithm in numerical analysis
The base case of the recursion could in principle be the sum of only one (or zero) numbers, but to amortize the overhead of recursion, one would normally
Kahan_summation_algorithm
Triangular array of the binomial coefficients
dth n-dimensional number. An alternative formula that does not involve recursion is P d ( n ) = 1 d ! ∏ k = 0 d − 1 ( n + k ) = n ( d ) d ! = ( n + d −
Pascal's_triangle
indices. Special feature allowing to match balanced constructs without recursion. Refers to the possibility of including quantifiers in look-behinds, thus
Comparison of regular expression engines
Comparison_of_regular_expression_engines
more advanced concepts that are occasionally needed: concurrency and recursion. Hierarchy is the most fundamental of all of the Warnier/Orr constructs
Warnier/Orr_diagram
French polymath (1623–1662)
triangle can also be represented: He defined the numbers in the triangle by recursion: Call the number in the (m + 1)th row and (n + 1)th column tmn. Then tmn = tm−1
Blaise_Pascal
Form of mathematical proof
Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and
Mathematical_induction
British-American psychological anthropologist (1904–1980)
doi:10.1002/sres.1110. Harries-Jones, P. (2004). "Revisiting Angels Fear: Recursion, Ecology and Aesthetics". Harries-Jones, P. (2005). "Gregory Bateson and
Gregory_Bateson
Three-holed sphere
"irreducible" surfaces among all surfaces of negative Euler characteristic. By a recursion argument, this implies that for any surface there is a system of simple
Pair_of_pants_(mathematics)
Figurate number
{\displaystyle S_{1}=1.} All square triangular numbers are found from the recursion S n = 34 S n − 1 − S n − 2 + 2 {\displaystyle S_{n}=34S_{n-1}-S_{n-2}+2}
Triangular_number
American activist and programmer (born 1953)
Laboratory. Until "around 1998", he maintained an office at the Institute that doubled as his legal residence. Stallman announced the plan for the GNU operating
Richard_Stallman
Language of the Basque people
inflected for the noun again. It has been estimated that with two levels of recursion, a Basque noun may have 458,683 inflected forms. The common noun liburu
Basque_language
Statistical model for 2D crystals
dislocations) is now covered in a renormalized (reduced) elasticity. The recursion relation for elasticity and fugacity are: d Y − 1 ( l ) d l = 3 2 π y
KTHNY_theory
Member of the ALGOL family of computer programming languages
chemistry of the 13 was excellent." The language originally did not include recursion. It was inserted into the specification at the last minute, against the
ALGOL_60
Computer programming language
Higher-order functions Generators Cooperative threads (coroutines) Tail recursion Exception handling Automatic memory management (mainly reference counting
Squirrel (programming language)
Squirrel_(programming_language)
Genus of Asian apes
self-embedded vocal motifs in wild orangutans, and the selective evolution of recursion". Annals of the New York Academy of Sciences. 1549 (1): 219–229. doi:10
Orangutan
Arithmetic operation
natural tetration function.) The proof is much the same as before; the recursion equation ensures that f ′ ( − 1 + ) = f ′ ( 0 + ) , {\displaystyle f^{\prime
Tetration
Algorithm for finding zeros of functions
equation in one variable has a p-adic root is Hensel's lemma, which uses the recursion from Newton's method on the p-adic numbers. Because of the more stable
Newton's_method
2014 edition of the C++ programming language standard
definitions must be available to the translation unit that uses them. Recursion can be used with a function of this type, but the recursive call must
C++14
Method of notation of very large integers
and if we start from n = 1 even for 3 ≤ b ≤ 9,999,999,999. Primitive recursion Cutler's bar notation Steinhaus–Moser notation For more details, see Powers
Knuth's_up-arrow_notation
Criticism of the Java programming language and Java software platform
pointers and recursion as the two gatekeeper concepts? Because he found them difficult? As Tim Bray points out, Java is perfectly adept at recursion, and concurrency
Criticism_of_Java
Numerical technique
one-dimensional adaptive quadratures to multi-dimensional integrals. On each recursion step the integral and the error are estimated using a plain Monte Carlo
Monte_Carlo_integration
DOUBLE RECURSION
DOUBLE RECURSION
Boy/Male
British, Christian, English
Dark Water; In the Seventeenth Century; Diminutive of Douglas
Boy/Male
Hindu, Indian
Money; Russian Currency
Surname or Lastname
French
French : from a reduced form of the Germanic personal name Hildo (see Hildebrand, Houde).French : habitational name from any of several places in Normandy called La Houle or Les Houles, named in Old French with the singular or plural of houle ‘cave’.English : variant of Hole.
Girl/Female
Scottish
From the Gaelic 'dubhglas' meaning dark water, dark stream, or from the dark river.
Surname or Lastname
English
English : possibly a variant of Goble or Gobel.Perhaps an Americanized spelling of French Gobeil.
Girl/Female
Christian, Hindu, Indian, Kannada
Money
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : variant of Double.In some cases, probably an altered spelling of South German Dobel or Döbel, a topographic name for someone who lived in a gorge or deep valley, Middle High German southern dialect tobel.
Surname or Lastname
English
English : of uncertain origin; perhaps derived from the vocabulary word soul as a term of affection.French (Soulé) : variant of Soulier 1.George Soule (1600–80), one of the passengers on the Mayflower in 1620, was one of the founders of Duxbury, MA, where he became comparatively wealthy. He left eight children.
Girl/Female
Latin
Mistress of the home.
Surname or Lastname
English
English : from a variant of the medieval personal name Tebald, Tibalt (see Theobald).
Surname or Lastname
English
English : variant of Coble.Americanized spelling of German Kobel.
Surname or Lastname
English
English : variant spelling of Dowdell.Possibly an altered spelling of German Daudel, Dautel, variants of Dietz.
Male
English
English name derived from the vocabulary word, from Latin nobilis, NOBLE means "noble."
Surname or Lastname
English
English : habitational name from Wombwell in South Yorkshire, named with the Old English byname Wamba meaning ‘belly’ (or this word used in a transferred topographical sense) + Old English well(a) ‘spring’, ‘stream’.
Boy/Male
Hindu
Born during the rainy season, Money
Male
English
Anglicized form of Irish Gaelic Dubhghall, DOYLE means "black stranger."Â
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : nickname from Old French doubel ‘twin’ (literally ‘double’, from Late Latin duplus, classical Latin duplex, from du(o) ‘two’ + plek, a root meaning ‘fold’).
Surname or Lastname
English
English : from the medieval personal name Dobbe, one of several pet forms of Robert in which the initial letter was altered. Compare Hobbs.
Surname or Lastname
English
English : variant of Dibble.Altered spelling of German Deibel or Deubel.
Surname or Lastname
English
English : nickname for a sickly person, from French debile ‘frail’, ‘weak’ (from Latin debilis).Americanized spelling of German Diebel.
DOUBLE RECURSION
DOUBLE RECURSION
Girl/Female
Arabic, Australian, Danish, French, German, Hebrew
Princess; Desert
Girl/Female
Christian, Hindu, Indian
Happiness
Boy/Male
Tamil
Ajathasathru | அஜாதாஷதà¯à®°à¯
Person who has no enemies
Boy/Male
Indian
Praise, Lauding
Girl/Female
Tamil
Neshwari | நேஷà¯à®µà®¾à®°à¯€
Neshwari is another name of Goddess Gayatri
Boy/Male
Hindu
Born with beauty
Boy/Male
Hindu
Representative of God, A type of a demi God
Boy/Male
Arabic, Australian, Muslim, Turkish
Follower of a Spiritual Path; Passable; Unobstructed
Boy/Male
Indian
Bringer of glad tidings, Human being
Biblical
asked or lent of God
DOUBLE RECURSION
DOUBLE RECURSION
DOUBLE RECURSION
DOUBLE RECURSION
DOUBLE RECURSION
n.
One who, or that which, doubles.
v. i.
To set up a word or words a second time by mistake; to make a doublet.
a.
To be the double of; to exceed by twofold; to contain or be worth twice as much as.
v. t.
To double the natural darkness of (a place).
imp. & p. p.
of Double
adv.
Twice; doubly.
v. t.
To load with a double charge, as of gunpowder.
n.
That which is doubled over or together; a doubling; a plait; a fold.
a.
To increase by adding an equal number, quantity, length, value, or the like; multiply by two; to double a sum of money; to double a number, or length.
adv.
In a double degree; doubly.
n.
A game between two pairs of players; as, a first prize for doubles.
n.
The act of one that doubles; a making double; reduplication; also, that which is doubled.
n.
Among compositors, a doublet (see Doublet, 2.); among pressmen, a sheet that is twice pulled, and blurred.
n.
Double-quick time, step, or march.
n.
The state of being double or doubled.
n.
Double beer; strong beer.
a.
Double; doubled; reduplicative; repeated.
adv.
In twice the quantity; to twice the degree; as, doubly wise or good; to be doubly sensible of an obligation.