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Let ( G m ) m ≥ 0 be an integer linear recurrence sequence (under some weak technical conditions) and let x ≥ 1 be an integer. In this paper, we are interested in the problem of finding combinations of the form x G n + G n − 1 which belongs to ( G m ) m ≥ 0 for infinitely many positive integers n . In this case, we shall make explicit an upper bound for x which only depends on the roots of the characteristic polynomial of this recurrence. As application, we shall study the k -nacci case. [ABSTRACT FROM AUTHOR]

• Convolutional neural networks and long short-term memory models are developed to train the model to recognize different human activities. • A convolutional neural network extract features from the input image, and PCA is used to extract deep features from the dataset. • Deep learning classifiers are used to attain better accuracy, and a comparison table is created to understand the improvement in the accuracy score. Human activity recognition (HAR) is an emerging field that identifies human actions in different settings. This activity is recognized by sensors placed in the room or residence where we wish to observe human action. Real-world applications and automation employ activity recognition to detect anomalous behavior. For example, the anomalous behavior of patients such as walking while advised to rest in bed and falling elderly people need to be monitored carefully in hospitals as well as in home-based monitoring systems. Security, healthcare, human interaction, and computer vision use it. The activity is monitored through sensors and cameras. There is no general, explicit approach for inferring human activities from sensor data. Sensor data and heuristics present technological challenges. Several elements must be evaluated to build a reliable activity recognition system. Factors such as storage, connectivity, processing, energy efficiency, and system adaptability are important. Deep learning systems can better recognize human activities from earlier datasets. In this study, the hybrid One Dimensional Convolution Neural Network with Long Short Term Memory (LSTM) classifier is employed to improve the performance of HAR. It offers a method for automatically and data-adaptively removing reliable characteristics from raw data. This model proposes a two-way classification for abstract and individual activity monitoring. Human activities such as walking, sitting, walking downstairs, walking upstairs, laying, and standing along with mobile phone usage are considered in this study. We also compare state-of-the-art algorithms such as Support Vector Machine (SVM), K-Nearest Neighbour (KNN), Long Short Term Memory (LSTM), and Convolutional Neural Network (CNN). The UCI-HAR dataset is used for recognizing human activity in the proposed work. Features such as mean, median, and autoregressive coefficients are derived from the raw data and processed with principal component analysis to make them more reliable. The LSTM model accepts a series of activities, whereas the CNN accepts a single input. The CNN takes the single input data and each of the outputs is forwarded to the LSTM model, which classifies the activity. The Hybrid model achieves 97.89% accuracy with the new feature selection methods, whereas the CNN and LSTM individually produce 92.77% and 92.80% accuracy. [ABSTRACT FROM AUTHOR]

For k ≥ 2, the k -generalized Fibonacci sequence ( F n ( k ) ) n is defined by the initial values 0 , 0 , … , 0 , 1 ( k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we shall study for which x, k and t the expression x t F n + t ( k ) + ⋯ + x F n + 1 ( k ) + F n ( k ) belongs to ( F m ( k ) ) m for infinitely many integers n . This work generalizes [13, Theorem 2] which is related to the case t = 1 . [ABSTRACT FROM AUTHOR]

• Introduction the Hofstadter-Möbius sequence. • Deserts in the Hofstadter-Möbius sequence. • Valleys in the Hofstadter-Möbius sequence. • Peaks in the Hofstadter-Möbius sequence. The Hofstadter Q -sequence is perhaps the most known example of meta-Fibonacci sequence. Many authors have been interested in meta-Fibonacci sequences related to this sequence. For example, recently, it was studied by A. Alkan, N. Fox, and O. Aybar the connection between the Q -sequence and the Hofstadter-Conway $ 10,000 sequence. Here, in the similar spirit as their paper, we study the interplay between the Hofstadter Q -sequence and one of the more important multiplicative arithmetic function, namely, the Möbius function. [ABSTRACT FROM AUTHOR]