Deriving Formulas for Integer Sequences Using Inductive Programming

2019 ◽  
pp. 16-24
Author(s):  
Les De Ridder ◽  
Thijs Vercammen
Keyword(s):  
Integer Sequences ◽  
Inductive Programming

INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1841-1853 ◽  
Author(s):  
B. K. MORIYA ◽  
C. J. SMYTH
Keyword(s):  
Prime Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Survey Methods ◽  
Integer Sequences ◽  
Positive Integers

We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is prime, complementing a result of D. H. Lehmer, who found the least n for which |τ(n)| is prime.

Constructions of QC LDPC codes based on integer sequences

2014 ◽  
Vol 57 (6) ◽  
pp. 1-14 ◽  
Author(s):  
LiJun Zhang ◽  
Bing Li ◽  
LeeLung Cheng
Keyword(s):  
Ldpc Codes ◽  
Integer Sequences

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

scholarly journals On the Number of Distributive Lattices

10.37236/1641 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Marcel Erné ◽  
Jobst Heitzig ◽  
Jürgen Reinhold
Keyword(s):  
Combinatorial Identities ◽  
Distributive Lattices ◽  
Integer Sequences ◽  
Isomorphism Classes ◽  
Exponential Bounds

We investigate the numbers $d_k$ of all (isomorphism classes of) distributive lattices with $k$ elements, or, equivalently, of (unlabeled) posets with $k$ antichains. Closely related and useful for combinatorial identities and inequalities are the numbers $v_k$ of vertically indecomposable distributive lattices of size $k$. We present the explicit values of the numbers $d_k$ and $v_k$ for $k < 50$ and prove the following exponential bounds: $$ 1.67^k < v_k < 2.33^k\;\;\; {\rm and}\;\;\; 1.84^k < d_k < 2.39^k\;(k\ge k_0).$$ Important tools are (i) an algorithm coding all unlabeled distributive lattices of height $n$ and size $k$ by certain integer sequences $0=z_1\le\cdots\le z_n\le k-2$, and (ii) a "canonical 2-decomposition" of ordinally indecomposable posets into "2-indecomposable" canonical summands.

scholarly journals On some new results for the generalised Lucas sequences

2021 ◽  
Vol 29 (1) ◽  
pp. 17-36
Author(s):  
Dorin Andrica ◽  
Ovidiu Bagdasar ◽  
George Cătălin Ţurcaş
Keyword(s):  
Integer Sequences ◽  
Lucas Sequences ◽  
Lucas Sequence ◽  
Necessary And Sufficient

Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

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