Results 1  10
of
666
THE FIBONACCI SEQUENCE MODULO N
"... Let n be a positive integer. The Fibonacci sequence, when considered modulo n, must repeat. In this note we investigate the period of repetition and the related unsolved problem of finding the smallest Fibonacci number divisible by n. The results given here are similar to those of the simple problem ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
of the Fibonacci sequence modulo p is the order of an elenjent £ in a group to be defined in §1. This result will allow us to estimate the period of repetition and the least Fibonacci number divisible by n. Sections 2 and 3 contain the exact statements of these theorems; in §4, related topics are discussed. 1
Catalan Numbers Modulo 2^k
, 2010
"... In this paper, we develop a systematic tool to calculate the congruences of some combinatorial numbers involving n!. Using this tool, we reprove Kummer’s and Lucas’ theorems in a unique concept, and classify the congruences of the Catalan numbers cn (mod 64). To achieve the second goal, cn (mod 8) ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
) and cn (mod 16) are also classified. Through the approach of these three congruence problems, we develop several general properties. For instance, a general formula with powers of 2 and 5 can evaluate cn (mod 2 k) for any k. An equivalence cn ≡ 2 k c¯n is derived, where ¯n is the number obtained
FIBONACCI SEQUENCES MODULO M
"... Most of the questions concerning the length of the period of the recurring sequence obtained by reducing a general Fibonacci sequence by a modulus m have been answered by D. D. Wall [ l]. The problem discussed in this paper is to determine the number of ordered pairs (a,b) with 0 ^ a < m and 0 ^ ..."
Abstract
 Add to MetaCart
it is reduced modulo m, taking least nonnegative residues. When h does not depend on a and b we may write h = h(m) instead. The special Fibonacci s equence which starts with the pair (0,1) will be denoted by {u} and its period when reduced modulo m by k(m). The sequence which starts with (2,1) will be denoted
Idempotents and nilpotents modulo n
"... Abstract. We study asymptotic properties of periods and transient phases associated with modular power sequences. The latter are simple; the former are vaguely related to the reciprocal sum of squarefree integer kernels. Let Zn denote the ring of integers modulo n. Define S(x) to be the sequence {x ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Abstract. We study asymptotic properties of periods and transient phases associated with modular power sequences. The latter are simple; the former are vaguely related to the reciprocal sum of squarefree integer kernels. Let Zn denote the ring of integers modulo n. Define S(x) to be the sequence
A Combinatorial proof of a recursive relation of the Motzkin sequence by Lattice Paths
 FIBONACCI QUART. 40 (2002), 3–8
, 2002
"... We consider those lattice paths in the Cartesian plane running from (05 0) that use the steps from S = {U = (1,1) (an upstep), I = (1,0) (a levelstep), D = (l,1) (a downstep)}. Let A(n, k) be the set of all lattice paths ending at the point (n, k) and let M(n) be the set of lattice paths in A(n9 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(n9 0) that never go below the xaxis. Let a(n9 k) \A(n9 k)  and mn \M(ri)\, where mn is called the Motzkin number. Here, we shall give a combinatorial proof of the threeterm recursion of the Motzkin sequence, and also that (n + 2)mn = (2w + V)mn_l + 3(/i l)mn_ 2? n + 2 m„ l m % nl • < 3 lim
Appendix: Periodic Behaviour of Some Recurrence Sequences Related to e, Modulo Powers of 2
, 709
"... Let the nth partial sum of the Taylor series e = ∑ ∞ r=0 1/r! be An/n!, and let pk/qk be the kth convergent of the simple continued fraction for e. Using a recent measure of irrationality for e, we prove weak versions of our conjecture that only two of the partial sums are convergents to e. We also ..."
Abstract
 Add to MetaCart
also show a surprising connection between the An and certain prime numbers, including 2, 5, 13, 37, and 463. In the Appendix, K. Schalm gives a conditional proof of the conjecture, assuming a certain other conjecture he makes about the An and qn modulo powers of 2. He presents tables supporting his
DISTRIBUTION OF POWERS MODULO 1 AND RELATED TOPICS
"... Abstract. This is a review of several results related to distribution of powers and combinations of powers modulo 1. We include a proof that given any sequence of real numbers θn, it is possible to get an α (given λ ̸ = 0),oraλ (given α>1) such that λαn is close to θn modulo1. Wealsoprovethatin a ..."
Abstract
 Add to MetaCart
Abstract. This is a review of several results related to distribution of powers and combinations of powers modulo 1. We include a proof that given any sequence of real numbers θn, it is possible to get an α (given λ ̸ = 0),oraλ (given α>1) such that λαn is close to θn modulo1. Wealsoprovethatin
Coil sensitivity encoding for fast MRI. In:
 Proceedings of the ISMRM 6th Annual Meeting,
, 1998
"... New theoretical and practical concepts are presented for considerably enhancing the performance of magnetic resonance imaging (MRI) by means of arrays of multiple receiver coils. Sensitivity encoding (SENSE) is based on the fact that receiver sensitivity generally has an encoding effect complementa ..."
Abstract

Cited by 193 (3 self)
 Add to MetaCart
separated pixel values for the originally superimposed positions. By repeating this procedure for each pixel in the reduced FOV a nonaliased fullFOV image is obtained. Unfolding is possible as long as the inversions in Eq. [2] can be performed. In particular, the number of pixels to be separated, n P
Hilly poor noncrossing partitions and (2, 3)Motzkin paths
"... Abstract. A hilly poor noncrossing partition is a noncrossing partition with the properties: (1) each block has at most two elements, (2) in its linear representation, any isolated vertex is covered by some arc. This paper defines basic pairs as a combinatorial object and gives the number of hilly p ..."
Abstract
 Add to MetaCart
poor noncrossing partitions with n blocks, which is closely related to Maximal DavenportSchinzel sequences. Authors introduce a class of generalized Motzkin paths called (i, j)Motzkin paths, and present a bijection between hilly poor noncrossing partitions and (2, 3)Motzkin paths. Specialization
Elliptic curves and related sequences
, 2003
"... A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coefficients λi and initial values are all 1. In this thesis we study the properties of EDSs and Somos 4 sequences reduced modulo a prime power pr. In chapter 2 we collect some results from number theory
Results 1  10
of
666