Integer sequences from geometry — lens sequences (original) (raw)
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arXiv:0710.3226, 2007
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious ``underground'' sequences underlying them are discovered in this paper.
The encyclopedia of integer sequences
1996
This paper gives a brief description of the author's database of integer sequences, now over 35 years old, together with a selection of a few of the most interesting sequences in the table. Many unsolved problems are mentioned. This paper was published (in a somewhat different form) in Sequences and their Applications
Sequences of Pentagonal Numbers
2013
Shyam Sunder Gupta has defined Smarandache consecutive and reversed Smarandache sequences of triangular numbers. Delfim F.M.Torres and Viorica Teca [1] have further investigated these sequences and defined mirror and symmetric Smarandache sequences of triangular numbers making use of Maple system. Working on the same lines we have defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of pentagonal numbers of dimension 2 using the Maple system .
2022
This note is devoted to study the recurrent numerical sequence defined by: a 0 = 0, a n = n 2 a n−1 + (n − 1)! (∀n ≥ 1). Although, it is immediate that (a n) n is constituted of rational numbers with denominators powers of 2, it is not trivial that (a n) n is actually an integer sequence. In this note, we prove this fact by expressing a n in terms of the Genocchi numbers and the Stirling numbers of the first kind. We derive from our main result several corollaries and we conclude with some remarks and open problems.
An Interesting Recursive Sequence - Part 2: Observations
Self (R.D. Heijnen), 2022
The Interesting Recursive Sequence comes down to a combination and intertwining of two Circles of Fifths, as known from music theory. One circle starts at “1” and is “going up” and the other one starts an octave higher at “2” and is “going down” where it mirrors what the first one does.
In this paper, we present a new integer sequence is developed from the recurrence relation) 0 J J with the initial conditions b J a, J 1 0 where a,b are not zeros simultaneously, is illustrated.
Pell–Padovan-circulant sequences and their applications
2017
This paper develops properties of recurrence sequences defined from circulant matrices obtained from the characteristic polynomial of the Pell–Padovan sequence. The study of these sequences modulo m yields cyclic groups and semigroups from the generating matrices. Finally, we obtain the lengths of the periods of the extended sequences in the extended triangle groups E(2, n, 2), E(2, 2, n) and E(n, 2, 2) for n ≥ 3 as applications of the results obtained.