Constant-recursive sequence explained

In mathematics, an infinite sequence of numbers

is called constant-recursive if it satisfies an equation of the form

$$s\_n\; =\; c\_1\; s\_\; +\; c\_2\; s\_\; +\; \backslash dots\; +\; c\_d\; s\_,$$for all

, where are constants. The equation is called a linear recurrence relation.The concept is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence.

For example, the Fibonacci sequence

,is constant-recursive because it satisfies the linear recurrence : each number in the sequence is the sum of the previous two. Other examples include the power of two sequence , where each number is the sum of twice the previous number, and the square number sequence . All arithmetic progressions, all geometric progressions, and all polynomials are constant-recursive. However, not all sequences are constant-recursive; for example, the factorial sequence is not constant-recursive.

Constant-recursive sequences are studied in combinatorics and the theory of finite differences. They also arise in algebraic number theory, due to the relation of the sequence to polynomial roots; in the analysis of algorithms, as the running time of simple recursive functions; and in the theory of formal languages, where they count strings up to a given length in a regular language. Constant-recursive sequences are closed under important mathematical operations such as term-wise addition, term-wise multiplication, and Cauchy product.

The Skolem–Mahler–Lech theorem states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. The Skolem problem, which asks for an algorithm to determine whether a linear recurrence has at least one zero, is an unsolved problem in mathematics.

Definition

A constant-recursive sequence is any sequence of integers, rational numbers, algebraic numbers, real numbers, or complex numbers

(written as as a shorthand) satisfying a formula of the form

$$s\_n\; =\; c\_1\; s\_\; +\; c\_2\; s\_\; +\; \backslash dots\; +\; c\_d\; s\_,$$

for all

for some fixed coefficients ranging over the same domain as the sequence (integers, rational numbers, algebraic numbers, real numbers, or complex numbers).The equation is called a linear recurrence with constant coefficients of order d.The order of the sequence is the smallest positive integer such that the sequence satisfies a recurrence of order d, or for the everywhere-zero sequence.

The definition above allows eventually-periodic sequences such as

and . Some authors require that , which excludes such sequences.^[1]

Examples

Selected examples of integer constant-recursive sequences^[2] ! Name !! Order (

d

  ) !! First few values !! Recurrence (for

n\ged

 ) !! Generating function !! OEIS

Zero sequence	0	0, 0, 0, 0, 0, 0, ...	sn=0	0 1
One sequence	1	1, 1, 1, 1, 1, 1, ...	sn=sn-1	1 1-x
Characteristic function of \{0\}	1	1, 0, 0, 0, 0, 0, ...	sn=0	1 1
	1	1, 2, 4, 8, 16, 32, ...	sn=2sn-1	1 1-2x
Powers of −1	1	1, −1, 1, −1, 1, −1, ...	sn=-sn-1	1 1+x
Characteristic function of \{1\}	2	0, 1, 0, 0, 0, 0, ...	sn=0	x 1
Decimal expansion of 1/6	2	1, 6, 6, 6, 6, 6, ...	sn=sn-1	1+5x 1-x
Decimal expansion of 1/11	2	0, 9, 0, 9, 0, 9, ...	sn=sn-2	9x 1-x2
	2	0, 1, 2, 3, 4, 5, ...	sn=2sn-1-sn-2	x (1-x)2
Odd positive integers	2	1, 3, 5, 7, 9, 11, ...	sn=2sn-1-sn-2	1+x (1-x)2
Fibonacci numbers	2	0, 1, 1, 2, 3, 5, 8, 13, ...	sn=sn-1+sn-2	x 1-x-x2
Lucas numbers	2	2, 1, 3, 4, 7, 11, 18, 29, ...	sn=sn-1+sn-2	2-x 1-x-x2
Pell numbers	2	0, 1, 2, 5, 12, 29, 70, ...	sn=2sn-1+sn-2	x 1-2x-x2
Powers of two interleaved with 0s	2	1, 0, 2, 0, 4, 0, 8, 0, ...	sn=2sn-2	1 1-2x2
	2	1, 1, 0, −1, −1, 0, 1, 1, ...	sn=sn-1-sn-2	1 1-x+x2
Triangular numbers	3	0, 1, 3, 6, 10, 15, 21, ...	sn=3sn-1-3sn-2+sn-3	x (1-x)3

Fibonacci and Lucas sequences

The sequence 0, 1, 1, 2, 3, 5, 8, 13, ... of Fibonacci numbers is constant-recursive of order 2 because it satisfies the recurrence

with . For example, and . The sequence 2, 1, 3, 4, 7, 11, ... of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions and . More generally, every Lucas sequence is constant-recursive of order 2.

Arithmetic progressions

For any

and any , the arithmetic progression is constant-recursive of order 2, because it satisfies . Generalizing this, see polynomial sequences below.

Geometric progressions

For any

and , the geometric progression is constant-recursive of order 1, because it satisfies . This includes, for example, the sequence 1, 2, 4, 8, 16, ... as well as the rational number sequence $1,\; \backslash frac12,\; \backslash frac14,\; \backslash frac18,\; \backslash frac,\; ...$.

Eventually periodic sequences

A sequence that is eventually periodic with period length

is constant-recursive, since it satisfies for all , where the order is the length of the initial segment including the first repeating block. Examples of such sequences are 1, 0, 0, 0, ... (order 1) and 1, 6, 6, 6, ... (order 2).

Polynomial sequences

A sequence defined by a polynomial

is constant-recursive. The sequence satisfies a recurrence of order (where is the degree of the polynomial), with coefficients given by the corresponding element of the binomial transform.^{[3] [4]} The first few such equations are for a degree 0 (that is, constant) polynomial, for a degree 1 or less polynomial, for a degree 2 or less polynomial, and for a degree 3 or less polynomial.

A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences.^[5] Any sequence of

integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order . If the initial conditions lie on a polynomial of degree or less, then the constant-recursive sequence also obeys a lower order equation.

Enumeration of words in a regular language

Let

be a regular language, and let be the number of words of length in . Then is constant-recursive. For example, for the language of all binary strings, for the language of all unary strings, and for the language of all binary strings that do not have two consecutive ones. More generally, any function accepted by a weighted automaton over the unary alphabet over the semiring (which is in fact a ring, and even a field) is constant-recursive.

Other examples

The sequences of Jacobsthal numbers, Padovan numbers, Pell numbers, and Perrin numbers are constant-recursive.

Non-examples

The factorial sequence

is not constant-recursive. More generally, every constant-recursive function is asymptotically bounded by an exponential function (see

1. Closed-form characterization

) and the factorial sequence grows faster than this. is not constant-recursive. This is because the generating function of the Catalan numbers is not a rational function (see

1. Equivalent definitions

).

Equivalent definitions

In terms of matrices

See main article: Companion matrix. |-style="text-align:center;"|

A sequence

is constant-recursive of order less than or equal to if and only if it can be written as where is a vector, is a matrix, and is a vector, where the elements come from the same domain (integers, rational numbers, algebraic numbers, real numbers, or complex numbers) as the original sequence. Specifically, can be taken to be the first values of the sequence, the linear transformation that computes from , and the vector .

In terms of non-homogeneous linear recurrences

|- class="wikitable"! Non-homogeneous !! Homogeneous|- align = "center"|

| |- align = "center"| |

A non-homogeneous linear recurrence is an equation of the form

where is an additional constant. Any sequence satisfying a non-homogeneous linear recurrence is constant-recursive. This is because subtracting the equation for from the equation for yields a homogeneous recurrence for , from which we can solve for to obtain

In terms of generating functions

|-style="text-align:center;"|

A sequence is constant-recursive precisely when its generating function

is a rational function , where and are polynomials and .Moreover, the order of the sequence is the minimum such that it has such a form with and .

The denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence:^[6]

where It follows from the above that the denominator must be a polynomial not divisible by (and in particular nonzero).

In terms of sequence spaces

|-align=center|

A sequence

is constant-recursive if and only if the set of sequences is contained in a sequence space (vector space of sequences) whose dimension is finite. That is, is contained in a finite-dimensional subspace of closed under the left-shift operator.

This characterization is because the order-

linear recurrence relation can be understood as a proof of linear dependence between the sequences for . An extension of this argument shows that the order of the sequence is equal to the dimension of the sequence space generated by for all .

Closed-form characterization

|-align=center|

}(1.618\ldots)^n - \frac(-0.618\ldots)^n

Constant-recursive sequences admit the following unique closed form characterization using exponential polynomials: every constant-recursive sequence can be written in the form

for all , where

• The term

is a sequence which is zero for all (where is the order of the sequence);

• The terms

are complex polynomials; and

• The terms

are distinct complex constants.

This characterization is exact: every sequence of complex numbers that can be written in the above form is constant-recursive.

For example, the Fibonacci number

is written in this form using Binet's formula: }\varphi^n - \frac\psi^n,where is the golden ratio and . These are the roots of the equation . In this case, , for all , are both constant polynomials, , and .

The term

is only needed when ; if then it corrects for the fact that some initial values may be exceptions to the general recurrence. In particular, for all .

The complex numbers

are the roots of the characteristic polynomial of the recurrence: whose coefficients are the same as those of the recurrence.We call the characteristic roots of the recurrence. If the sequence consists of integers or rational numbers, the roots will be algebraic numbers.If the roots are all distinct, then the polynomials are all constants, which can be determined from the initial values of the sequence.If the roots of the characteristic polynomial are not distinct, and is a root of multiplicity , then in the formula has degree . For instance, if the characteristic polynomial factors as , with the same root r occurring three times, then the th term is of the form ^[7]

Closure properties

Examples

The sum of two constant-recursive sequences is also constant-recursive. For example, the sum of

and is ( ), which satisfies the recurrence . The new recurrence can be found by adding the generating functions for each sequence.

Similarly, the product of two constant-recursive sequences is constant-recursive. For example, the product of

and is ( ), which satisfies the recurrence .

The left-shift sequence

and the right-shift sequence (with ) are constant-recursive because they satisfy the same recurrence relation. For example, because is constant-recursive, so is .

List of operations

In general, constant-recursive sequences are closed under the following operations, where

denote constant-recursive sequences, are their generating functions, and are their orders, respectively.

Order
Term-wise sum s+t	(s+t)n=sn+tn	—	f(x)+g(x)	\led+e
Term-wise product s ⋅ t	(s ⋅ t)n=sn ⋅ tn	—	1 2\pii \int\gamma f(\zeta) g\left( \zeta x \zeta \right) d\zeta [8] [9]	\led ⋅ e
Cauchy product s*t	(s*t)n= n \sum i=0 sitn-i	—	f(x)g(x)	\led+e
Left shift Ls	(Ls)n=sn+1	—	f(x)-s0 x	\led
Right shift Rs	(Rs)n=\begin{cases}sn-1&n\ge1\\0&n=0\end{cases}	—	xf(x)	\led+1
Cauchy inverse s(-1)	(s(-1))n= \sum {i1+...+ik=n \atop{i1,\ldots,ik\ne0}}(-1)k s i1 s i2 … s ik	s0=1	1 f(x)	\led+1
Kleene star s(*)	(s(*))n= \sum {i1+...+ik=n \atop{i1,\ldots,ik\ne0}} s i1 s i2 … s ik	s0=0	1 1-f(x)	\led+1

The closure under term-wise addition and multiplication follows from the closed-form characterization in terms of exponential polynomials. The closure under Cauchy product follows from the generating function characterization. The requirement

for Cauchy inverse is necessary for the case of integer sequences, but can be replaced by if the sequence is over any field (rational, algebraic, real, or complex numbers).

Behavior

See main article: Skolem–Mahler–Lech theorem and Skolem problem.

Zeros

Despite satisfying a simple local formula, a constant-recursive sequence can exhibit complicated global behavior. Define a zero of a constant-recursive sequence to be a nonnegative integer

such that . The Skolem–Mahler–Lech theorem states that the zeros of the sequence are eventually repeating: there exists constants and such that for all , if and only if . This result holds for a constant-recursive sequence over the complex numbers, or more generally, over any field of characteristic zero.^[10]

Decision problems

The pattern of zeros in a constant-recursive sequence can also be investigated from the perspective of computability theory. To do so, the description of the sequence

must be given a finite description; this can be done if the sequence is over the integers or rational numbers, or even over the algebraic numbers.^[11] Given such an encoding for sequences , the following problems can be studied:

Notable decision problems! Problem !! Description !! Status^[12]

Existence of a zero (Skolem problem)	On input (sn) infty n=0 , is sn=0 for some n ?	Open
Infinitely many zeros	On input (sn) infty n=0 , is sn=0 for infinitely many n ?	Decidable
Eventually all zero	On input (sn) infty n=0 , is sn=0 for all sufficiently large n ?	Decidable
Positivity	On input (sn) infty n=0 , is sn>0 for all n ?	Open
Eventual positivity	On input (sn) infty n=0 , is sn>0 for all sufficiently large n ?	Open

Because the square of a constant-recursive sequence

is still constant-recursive (see closure properties), the existence-of-a-zero problem in the table above reduces to positivity, and infinitely-many-zeros reduces to eventual positivity. Other problems also reduce to those in the above table: for example, whether for some reduces to existence-of-a-zero for the sequence . As a second example, for sequences in the real numbers, weak positivity (is for all ?) reduces to positivity of the sequence (because the answer must be negated, this is a Turing reduction).

The Skolem-Mahler-Lech theorem would provide answers to some of these questions, except that its proof is non-constructive. It states that for all

, the zeros are repeating; however, the value of is not known to be computable, so this does not lead to a solution to the existence-of-a-zero problem.^[11] On the other hand, the exact pattern which repeats after is computable.^{[11] [13]} This is why the infinitely-many-zeros problem is decidable: just determine if the infinitely-repeating pattern is empty.

Decidability results are known when the order of a sequence is restricted to be small. For example, the Skolem problem is decidable for algebraic sequences of order up to 4.^{[14] [15] [16]} It is also known to be decidable for reversible integer sequences up to order 7, that is, sequences that may be continued backwards in the integers.^[12]

Decidability results are also known under the assumption of certain unproven conjectures in number theory. For example, decidability is known for rational sequences of order up to 5 subject to the Skolem conjecture (also known as the exponential local-global principle). Decidability is also known for all simple rational sequences (those with simple characteristic polynomial) subject to the Skolem conjecture and the weak p-adic Schanuel conjecture.^[17]

Degeneracy

Let

be the characteristic roots of a constant recursive sequence . We say that the sequence is degenerate if any ratio is a root of unity, for . It is often easier to study non-degenerate sequences, in a certain sense one can reduce to this using the following theorem: if has order and is contained in a number field of degree over , then there is a constant

such that for some

each subsequence is either identically zero or non-degenerate.^[18]

Generalizations

A D-finite or holonomic sequence is a natural generalization where the coefficients of the recurrence are allowed to be polynomial functions of

rather than constants.^[19]

A

-regular sequence satisfies a linear recurrences with constant coefficients, but the recurrences take a different form. Rather than

being a linear combination of for some integers that are close to , each term in a -regular sequence is a linear combination of for some integers whose base- representations are close to that of .^[20] Constant-recursive sequences can be thought of as -regular sequences, where the base-1 representation of consists of copies of the digit .

References

• Book: Brousseau, Alfred . Linear Recursion and Fibonacci Sequences . 1971 . Fibonacci Association .
• Book: Kauers . Manuel . Paule . Peter . The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates . 2010 . Springer Vienna . 978-3-7091-0444-6 . en . 66 .
• Book: Stanley, Richard P. . 2011 . Enumerative Combinatorics . 1 . 2 . Cambridge studies in advanced mathematics.

External links

• Web site: OEIS Index Rec. OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)

Notes and References

1. Halava. Vesa. Skolem's Problem – On the Border between Decidability and Undecidability. 2005. Harju. Tero. Hirvensalo. Mika. Karhumäki. Juhani. 10.1.1.155.2606. 1.
2. Web site: Index to OEIS: Section Rec - OeisWiki . 2024-04-18 . oeis.org.
3. Boyadzhiev. Boyad. 2012. Close Encounters with the Stirling Numbers of the Second Kind. Math. Mag.. 85. 4. 252–266. 10.4169/math.mag.85.4.252. 1806.09468. 115176876.
4. Riordan . John . 1964 . Inverse Relations and Combinatorial Identities . The American Mathematical Monthly . en . 71 . 5 . 485–498 . 10.1080/00029890.1964.11992269 . 0002-9890.
5. Book: Jordan. Charles. Calculus of Finite Differences. Jordán. Károly. 1965. American Mathematical Soc.. 978-0-8284-0033-6. en. 9–11. See formula on p.9, top.
6. On the variety of linear recurrences and numerical semigroups. Semigroup Forum. 2013-11-14. 0037-1912. 569–574. 88. 3. 10.1007/s00233-013-9551-2. en. Ivan. Martino. Luca. Martino. 1207.0111. 119625519.
7. Book: 2.1.1 Constant coefficients – A) Homogeneous equations. Mathematics for the Analysis of Algorithms. Daniel H.. Greene. Donald E.. Knuth. Donald Knuth. 2nd. Birkhäuser . 17 . 1982. .
8. Pohlen . Timo . 2009 . The Hadamard product and universal power series . 36–37 . University of Trier (Doctoral Dissertation).
9. See Hadamard product (series) and Parseval's theorem.
10. Lech. C.. A Note on Recurring Series. Arkiv för Matematik. 2. 417–421. 1953. 5. 10.1007/bf02590997. 1953ArM.....2..417L . free.
11. Book: Ouaknine . Joël . Worrell . James . Decision problems for linear recurrence sequences . 10.1007/978-3-642-33512-9_3 . 3040104 . 21–28 . Springer-Verlag . Heidelberg . Lecture Notes in Computer Science . Reachability Problems: 6th International Workshop, RP 2012, Bordeaux, France, September 17–19, 2012, Proceedings . 7550 . 2012. 978-3-642-33511-2 . .
12. Book: Lipton . Richard . Luca . Florian . Nieuwveld . Joris . Ouaknine . Joël . Purser . David . Worrell . James . On the Skolem Problem and the Skolem Conjecture . 2022-08-04 . Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science . https://dl.acm.org/doi/10.1145/3531130.3533328 . LICS '22 . New York, NY, USA . Association for Computing Machinery . 1–9 . 10.1145/3531130.3533328 . 978-1-4503-9351-5.
13. Berstel. Jean. Mignotte. Maurice. 1976. Deux propriétés décidables des suites récurrentes linéaires. Bulletin de la Société Mathématique de France. fr. 104. 175–184. 10.24033/bsmf.1823. free.
14. Vereshchagin . N. K. . 1985-08-01 . Occurrence of zero in a linear recursive sequence . Mathematical Notes of the Academy of Sciences of the USSR . en . 38 . 2 . 609–615 . 10.1007/BF01156238 . 1573-8876.
15. Tijdeman . R. . Mignotte . M. . Shorey . T. N. . 1984 . The distance between terms of an algebraic recurrence sequence. . Journal für die reine und angewandte Mathematik . 349 . 63–76 . 0075-4102.
16. Bacik . Piotr . Completing the picture for the Skolem Problem on order-4 linear recurrence sequences . 2024-09-02 . cs.FL . 2409.01221.
17. Bilu . Yuri . Skolem Meets Schanuel . 2022-04-28 . 2204.13417 . Luca . Florian . Nieuwveld . Joris . Ouaknine . Joël . Purser . David . Worrell . James. cs.LO .
18. Book: Recurrence sequences . 2003 . American Mathematical Society . 978-0-8218-3387-2 . Everest . Graham . Mathematical surveys and monographs . Providence, RI . 5.
19. Stanley . Richard P . 1980 . Differentiably finite power series . European Journal of Combinatorics . 1 . 2 . 175–188. 10.1016/S0195-6698(80)80051-5 .
20. Allouche . Jean-Paul . Shallit . Jeffrey . 1992 . The ring of k-regular sequences . Theoretical Computer Science . 98 . 2 . 163–197. 10.1016/0304-3975(92)90001-V .