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Infinite sequence of numbers satisfying a linear equation
an infinite sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is called constant-recursive if it satisfies
Constant-recursive_sequence
Unsolved problem in mathematics
problem in mathematics Is there an algorithm to test whether a constant-recursive sequence has a zero? More unsolved problems in mathematics In mathematics
Skolem_problem
Ordered list of whole numbers
Thue–Morse sequence Ulam numbers Weird numbers Wolstenholme number Constant-recursive sequence On-Line Encyclopedia of Integer Sequences List of integer
Integer_sequence
Finite or infinite ordered list of elements
sequence Thue–Morse sequence List of integer sequences Types ±1-sequence Arithmetic progression Automatic sequence Cauchy sequence Constant-recursive
Sequence
Base of natural logarithms
The number e is a mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352
E_(mathematical_constant)
Pattern defining an infinite sequence of numbers
"Linear recursive sequences". SIAM Rev. Vol. 10, no. 3. pp. 324–353. JSTOR 2027658. Brousseau, Alfred (1971). Linear Recursion and Fibonacci Sequences. Fibonacci
Recurrence_relation
Use of functions that call themselves
solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own
Recursion_(computer_science)
Iterative algorithm on numbers
the sequence of x 3-digit constants 495 (2) n = 4 + 2 x ( x ≥ 0 ) , {\displaystyle n=4+2x\quad (x\geq 0)\,,} ...... Sequence of 4-digit constant 6174
Kaprekar's_routine
nonnegative integer appear in Recamán's sequence? Skolem problem: can an algorithm determine if a constant-recursive sequence contains a zero? The values of g(k)
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Process of repeating items in a self-similar way
One's parent (base case), or One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as
Recursion
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Transcendental number(s) with all positive integers in order
base) in some recursive order. For instance, the binary Champernowne sequence in shortlex order is 0 1 00 01 10 11 000 001 ... (sequence A076478 in the
Champernowne_constant
Infinite sequence in mathematics
other scales. Bertran Steinsky has created a recursive formula for the i-th term of the sequence. The sequence is not eventually periodic, that is, its terms
Kolakoski_sequence
Mathematical relation defining a sequence
combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Quickly growing function
computable functions are primitive recursive. It is essentially constructed by diagonalizing a sequence of primitive recursive functions f1,f2,… selected from
Ackermann_function
Numbers obtained by adding the two previous ones
S2CID 35413237. Pethő, Attila (2001). "Diophantine properties of linear recursive sequences II". Acta Mathematica Academiae Paedagogicae Nyíregyháziensis. 17:
Fibonacci_sequence
Number sequence 3,0,2,3,2,5,5,7,10,...
In mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation x3 = x + 1. The Perrin numbers
Perrin_number
Type of functions, in mathematical analysis
the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they
Holonomic_function
Linear recurrence equation
mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations
P-recursive_equation
Certain constant-recursive integer sequences
Lucas sequences U n ( P , Q ) {\displaystyle U_{n}(P,Q)} and V n ( P , Q ) {\displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that
Lucas_sequence
Fractal named after mathematician Benoit Mandelbrot
recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a fractal curve. The "style" of this recursive detail
Mandelbrot_set
Exponentation in modular arithmetic
pages follow describing how such sequences might be contrived in general. The m-th term of any constant-recursive sequence (such as Fibonacci numbers or
Modular_exponentiation
Integer sequence
van de Pol. The sequence can be broken into discrete "block" and "glue" sequences, which can be used to recursively build up the sequence. For example,
Gijswijt's_sequence
Halting probability of a random computer program
binary sequence representing the real number is an algorithmically random sequence. Calude, Hertling, Khoussainov, and Wang showed that a recursively enumerable
Chaitin's_constant
Infinite binary sequence generated by repeated complementation and concatenation
resulting algorithm takes constant time to generate each sequence element, using only a logarithmic number of bits (constant number of words) of memory
Thue–Morse_sequence
Subroutine call performed as final action of a procedure
target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end
Tail_call
Mathematical function that can be computed by a program
be simply constants. A subset of these is the primitive recursive functions. Another example is the Ackermann function, which is recursively defined but
Computable_function
Product of numbers from 1 to n
multiplying the numbers from 1 to n in sequence is inefficient, because it involves n multiplications, a constant fraction of which take time O(nlog2n)
Factorial
The zeros of a linear recurrence relation mostly form a regularly repeating pattern
n ) {\displaystyle (u_{n})} be a constant-recursive sequence with values in K {\displaystyle K} , i.e., a sequence satisfying a recurrence relation of
Skolem–Mahler–Lech_theorem
Divide and conquer sorting algorithm
sequences are then partitioned into r {\displaystyle r} parts and assigned to the appropriate processor groups. These steps are repeated recursively in
Merge_sort
Large number coined by Ronald Graham
computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Ronald Graham, the number's namesake. As there is a recursive formula
Graham's_number
Cycle through all length-k sequences
recursively constructed sequences and extend to the two-dimensional case. de Bruijn decoding is of interest, e.g., in cases where large sequences or
De_Bruijn_sequence
Mathematical sequences
In mathematics, the Fibonacci numbers form a sequence defined recursively by: F n = { 0 n = 0 1 n = 1 F n − 1 + F n − 2 n > 1 {\displaystyle
Generalizations of Fibonacci numbers
Generalizations_of_Fibonacci_numbers
Problem optimization method
new sub-problems. For example, consider the recursive formulation for generating the Fibonacci sequence: Fi = Fi−1 + Fi−2, with base case F1 = F2 = 1
Dynamic_programming
Algorithm for fast exponentiation
function In each recursive call, the least-significant digit of the binary representation of n is removed. It follows that the number of recursive calls is ⌈
Exponentiation_by_squaring
Generalization of addition, multiplication, exponentiation, tetration, etc.
hyperoperation sequence as a whole is seen to be a version of the original Ackermann function ϕ ( a , b , n ) {\displaystyle \phi (a,b,n)} — recursive but not
Hyperoperation
Generates a forecast of future values of a time series
Poisson's use of recursive exponential window functions in convolutions from the 19th century, as well as Kolmogorov and Zurbenko's use of recursive moving averages
Exponential_smoothing
Type of mathematical sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N {\displaystyle N} , its subsequence x 1 , … , x N {\displaystyle
Low-discrepancy_sequence
Recursive integer sequence
the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named
Catalan_number
Any of several recursively-defined integer sequences
background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are: The Hofstadter Figure-Figure
Hofstadter_sequence
Infinite sequence in mathematics
{\displaystyle t_{n}} in the regular paperfolding sequence, starting with n = 1 {\displaystyle n=1} , can be found recursively as follows. Divide n {\displaystyle n}
Regular_paperfolding_sequence
Infinite integer series where the next number is the sum of the two preceding it
complementary instances of Lucas sequences. The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of
Lucas_number
Limit of a uniformly computable sequence of functions
limit of a uniformly computable sequence of functions. The terms computable in the limit, limit recursive and recursively approximable are also used. One
Computation_in_the_limit
Programming language
simple register language designed to precisely capture the primitive recursive functions. The language is derived from the counter-machine model. Like
LOOP_(programming_language)
Unique positive real number which when multiplied by itself gives 2
for Mathematical Constants (PDF). Archived (PDF) from the original on 2011-06-10. Retrieved 2010-04-30. Sloane, N. J. A. (ed.). "Sequence A082405 (a(n) =
Square_root_of_2
Sequence of rational numbers
ISBN 978-1-4757-1740-2. Stone, Alex (2023). "The Astonishing Behavior of Recursive Sequences". Quanta Magazine. Retrieved 2023-11-17. Matsuhira, Rinnosuke; Matsusaka
Göbel's_sequence
Arithmetic operation
not an elementary recursive function. One can prove by induction that for every elementary recursive function f, there is a constant c such that f ( x
Tetration
Rudin–Shapiro sequence. However, the Rudin–Shapiro sequence cannot be expressed as the fixed point of some uniform morphism alone. There is a recursive definition
Rudin–Shapiro_sequence
Technique for defining number-theoretic functions by recursion
primitive recursive function. In the context of primitive recursive functions, it is convenient to have a means to represent finite sequences of natural
Course-of-values_recursion
Formal power series
a sequence can be expressed as a rational function (the ratio of two finite-degree polynomials) if and only if the sequence is a linear recursive sequence
Generating_function
Control technique for improving qubit coherence in quantum computing
Carr-Purcell-Meiboom-Gill (CPMG) sequence to more advanced, non-periodic sequences like Uhrig Dynamical Decoupling (UDD) and recursive, high-order schemes like
Dynamical_decoupling
Algorithmic technique
In particular, pairwise summation of a sequence of n numbers xn works by recursively breaking the sequence into two halves, summing each half, and adding
Pairwise_summation
Measure of unsolvability
⟨ ≤, = ⟩. A degree is called recursively enumerable (r.e.) or computably enumerable (c.e.) if it contains a recursively enumerable set. Every r.e. degree
Turing_degree
Study of computable functions and Turing degrees
mathematical constructions can be effectively performed is sometimes called recursive mathematics. Computability theory originated in the 1930s, with the work
Computability_theory
Numbers significantly larger than those used regularly
after the double arrow is a very large number itself, the above can recursively be applied to that value. Examples: 10↑↑1010103.81×1017 (between 10↑↑↑2
Large_numbers
Subfield of information theory and computer science
to an additive constant depending only on the choice of universal Turing machine. For this reason the set of random infinite sequences is independent
Algorithmic information theory
Algorithmic_information_theory
Tree node with two other nodes as descendants
questions can be answered in constant time. Hence, case 1 can be answered in linear space and constant time. The sequence of RMQ that reduced from LCA
Lowest_common_ancestor
Mathematical puzzle game
clockwise. It suffices to represent the sequence of disks to be moved. The solution can be found using two mutually recursive procedures: To move n disks counterclockwise
Tower_of_Hanoi
Describes approximate behavior of a function
Master theorem (analysis of algorithms): For analyzing divide-and-conquer recursive algorithms using big O notation Nachbin's theorem: A precise method of
Big_O_notation
Mathematical constant
of the uniqueness of the Foias constant may also be applied to other similar recursive sequences. Mathematical constant Ewing, J. and Foias, C. "An Interesting
Foias_constant
Thesis on the nature of computability
with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments)
Church–Turing_thesis
Fundamental theorem in mathematical logic
interpret its own construction, so that this construction is non-recursive (as recursive definitions would be unambiguous). Also, if T {\displaystyle T}
Gödel's_completeness_theorem
Abstract model of computation
primitive recursive functions ( cf Minsky (1967), Boolos-Burgess-Jeffrey (2002) ). (How to cast the net wider to capture the total and partial mu recursive functions
Random-access_machine
Type of grammar for describing formal languages
closer to how string recognition tends to be done in practice, e.g. by a recursive descent parser. Unlike CFGs, PEGs cannot be ambiguous; a string has exactly
Parsing_expression_grammar
Astonishing Behavior of Recursive Sequences", Quanta Magazine Hone, Andrew N. W. (2005), "Elliptic Curves and Quadratic Recurrence Sequences", Bulletin of the
Somos_sequence
Sequence of integers
This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the golden ratio does to the Fibonacci sequence. P(n) is
Padovan_sequence
Abstract machine used in a formal logic and theoretical computer science
equivalence, capable of creating the total- and partial- recursive functions: Zero function (or constant function) Successor function Identity function Composition
Counter_machine
Real number that can be computed within arbitrary precision
terminating algorithm. They are also known as the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number
Computable_number
Mathematical-logic system
is M; this means a recursive function definition cannot be written with let. The letrec construction would allow writing recursive function definitions
Lambda_calculus
Divide and conquer sorting algorithm
only constant additional space before making any recursive call. Quicksort must store a constant amount of information for each nested recursive call
Quicksort
Computer science professor
demonstrating that a recursively enumerable real number is an algorithmically random sequence if and only if it is a Chaitin's constant for some encoding
Yongge_Wang
Sum of an infinite series, about 0.6434
} (sequence A118227 in the OEIS). Here ( s i ) i ≥ 0 {\displaystyle (s_{i})_{i\geq 0}} denotes Sylvester's sequence, which is defined recursively by s
Cahen's_constant
Computer science and recursion theory
of recursive functions by use of the IF-THEN-ELSE construction common to computer science, together with four of the operators of primitive recursive functions:
McCarthy_Formalism
Problem in computer science
{\displaystyle \alpha } , there is a 1-1 total recursive function f {\displaystyle f} and a constant c {\displaystyle c} such that for all i {\displaystyle
Halting_problem
Sequence of numbers with distinct sums
In mathematics, the Mian–Chowla sequence is an integer sequence defined recursively in the following way. The sequence starts with a 1 = 1. {\displaystyle
Mian–Chowla_sequence
Non-comparative lexicographical sorting algorithm
The 0s bin and the 1s bin are then sorted recursively based on the next bit of each array element. Recursive processing continues until the least significant
Radix_sort
Value approached by a mathematical object
X=ℝ and S=[0,1]∪{2}. Then 2∈S, and therefore is the limit of the constant sequence 2,2,⋯. But 2 is not a limit point of S. A closed set, which is defined
Limit_(mathematics)
The Ehrenfeucht–Mycielski sequence is a recursively defined sequence of binary digits with pseudorandom properties, defined by Andrzej Ehrenfeucht and
Ehrenfeucht–Mycielski sequence
Ehrenfeucht–Mycielski_sequence
Class of artificial neural network
similarly the recursive neural network's backpropagation is called "backpropagation through structure", where the said structure is the DAG. Recursive neural
Recurrent_neural_network
Two functions defined from each other
this example, the mutually recursive calls are tail calls, and tail call optimization would be necessary to execute in constant stack space. In C, this would
Mutual_recursion
Binary tree derived from a sequence of numbers
from a sequence of distinct numbers. To construct the Cartesian tree, set its root to be the minimum number in the sequence, and recursively construct
Cartesian_tree
Mathematical logic concept
strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence. Gentzen showed that the consistency
Gentzen's_consistency_proof
Counts pieces of a disk cut by lines
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza
Lazy_caterer's_sequence
Method of machine learning
learning is not possible, though a form of hybrid online learning with recursive algorithms can be used where ft+1 is permitted to depend on ft and all
Online_machine_learning
Sorting algorithm
clear a spot for x = A[i]. The algorithm can also be implemented in a recursive way. The recursion just replaces the outer loop, calling itself and storing
Insertion_sort
String that is strictly smaller in lexicographic order than all of its rotations
generated in time proportional to the length of the sequence. A constant fraction of the words in this sequence have length exactly n {\displaystyle n} , so
Lyndon_word
Sequence of words formed by specific rules
the basis for a 1947 proof "that the word problem for semigroups was recursively insoluble", and later devised the canonical system for the creation of
Formal_language
Branch of mathematical logic
reverse mathematics. The initials "RCA" stand for "recursive comprehension axiom", where "recursive" means "computable", as in computable function. This
Reverse_mathematics
Number, approximately 1.618
where the integers Fn and Fn−1 are consecutive Fibonacci numbers. The recursive formula also leads to another property of the positive powers of φ;
Golden_ratio
Data flowchart
on. The doubled bracket indicates that the set is recursive. Data structures that are truly recursive are rather rare. One source mentioned Warnier-Orr
Warnier/Orr_diagram
Sequence of program instructions invokable by other software
hardware supports only a few levels of subroutine nesting, but can support recursive subroutines. Machines before the mid-1960s, such as the UNIVAC I, the
Function (computer programming)
Function_(computer_programming)
Algorithm to search the nodes of a graph
edges. The recursive implementation will visit the nodes from the example graph in the following order: A, B, D, F, E, C, G. The non-recursive implementation
Depth-first_search
Family of higher-order functions
function that analyzes a recursive data structure and, through use of a given combining operation, recombines the results of recursively processing its constituent
Fold_(higher-order_function)
Non-contradiction of a theory
strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable
Consistency
Proof in set theory
i.e. the set of counting numbers for the subcountable sets may not be recursive and can thus fail to be countable. The elaborate collection of subsets
Cantor's_diagonal_argument
The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational)
List_of_representations_of_e
Algorithm that arranges lists in order
"in-place". Recursion: Some algorithms are either typically recursive or typically non-recursive, while others may typically be both (e.g., merge sort). Stability:
Sorting_algorithm
Mathematical modeling language
and Schmitz whose algorithm admits a primitive-recursive upper bound when the dimension is a constant. The mutual reachability problem (aka reversible
Vector_addition_system
Type of mathematical function
numbers. A recursive ordinal notation must satisfy the following two additional properties: the subset of natural numbers is a recursive set the induced
Ordinal_notation
Combinatorial algorithm
sequence of permutations generated by the Steinhaus–Johnson–Trotter algorithm has a natural recursive structure, that can be generated by a recursive
Steinhaus–Johnson–Trotter algorithm
Steinhaus–Johnson–Trotter_algorithm
travel, tourism, insurance
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
Boy/Male
Tamil
Constant
Boy/Male
Latin English
Constant.
Boy/Male
Latin
Constant.
Surname or Lastname
English and French
English and French : from the medieval female personal name Constance, Latin Constantia, originally a feminine form of Constantius (see Constant), but later taken as the abstract noun constantia ‘steadfastness’.English and French : habitational name from Coutances in La Manche, France, which was named Constantia in Latin (see above) in honor of the Roman emperor Constantius Chlorus, who was responsible for fortifying the settlement in ad 305.
Girl/Female
Latin American English French Shakespearean
Firm of purpose. Constancy, from the Latin Constantia.
Boy/Male
Australian, British, Danish, English, French, German, Italian, Latin, Swedish, Swiss
Steadfast; Constant
Female
Spanish
Spanish form of Latin Constantia, CONSTANZA means "steadfast."
Girl/Female
Australian, German, Latin, Spanish, Swedish
Constancy; Steadfastness
Surname or Lastname
French and English
French and English : from a medieval personal name (Latin Constans, genitive Constantis, meaning ‘steadfast’, ‘faithful’, present participle of the verb constare ‘stand fast’, ‘be consistent’). This was borne by an 8th-century Irish martyr. This surname has also absorbed some cases of surnames based on Constantius, a derivative of Constans, borne by a 2nd-century martyr, bishop of Perugia. Compare Constantine.English : perhaps also a nickname from Old French constant ‘steadfast’, ‘faithful’.
Female
Romanian
Romanian form of Latin Constantia, CONSTANTA means "steadfast."
Girl/Female
Australian, French, German, Latin, Spanish
Constancy; Steadfastness
Girl/Female
Spanish Italian
Constant.
Boy/Male
Tamil
Nityagopal | நிதà¯à®¯à®•à¯‹à®ªà®¾à®²Â
Constant
Nityagopal | நிதà¯à®¯à®•à¯‹à®ªà®¾à®²Â
Girl/Female
British, English
Similar to Constance; Used by 16th and 17th Century Puritans
Male
French
French and Romanian form of Latin Constantinus, CONSTANTIN means "steadfast."Â
Female
English
English form of Latin Constantia, CONSTANCE means "steadfast."Â
Girl/Female
Latin English
Firm of purpose. Constancy, from the Latin Constantia.
Male
Polish
Polish form of Latin Constans, KONSTANTY means "steadfast."
Boy/Male
English Latin
Steady; stable.
Girl/Female
American, Australian, British, Christian, Dutch, English, French, German, Latin, Portuguese, Shakespearean, Swedish
Constancy; Steadfastness
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
CONSTANT RECURSIVE-SEQUENCE
n.
A character used in cursive writing.
a.
A day of the present or current month; as, the sixth instant; -- an elliptical expression equivalent to the sixth of the month instant, i. e., the current month. See Instant, a., 3.
adv.
With constancy; steadily; continually; perseveringly; without cessation; uniformly.
n.
The state or quality of being constant or steadfast; freedom from change; stability; fixedness; immutability; as, the constancy of God in his nature and attributes.
n.
A revulsive medicine.
adv.
In a decursive manner.
n.
That which causes revulsion; specifically (Med.), a revulsive remedy or agent.
a.
harmonizing together; accordant; as, consonant tones, consonant chords.
n.
An expression of assent to a bill or motion; an affirmative vote; also, a member who votes "Content.".
a.
Serving, or able, to repulse; repellent; as, a repulsive force.
a.
Repulsive; driving back.
a.
Not constant; not stable or uniform; subject to change of character, appearance, opinion, inclination, or purpose, etc.; not firm; unsteady; fickle; changeable; variable; -- said of persons or things; as, inconstant in love or friendship.
a.
Cold; forbidding; offensive; as, repulsive manners.
v. t.
Causing revulsion; revulsive.
a.
Not constant; inconstant; fickle; changeable.
adv.
Constant; continual.
n.
The quality or state of being inconstant; want of constancy; mutability; fickleness; variableness.
n.
A superior wine, white and red, from Constantia, in Cape Colony.
a.
Prone to make excursions; wandering; roving; exploring; as, an excursive fancy.
n.
A constant irritating desire.
travel, tourism, insurance