Linear sequences with constant coefficients
In particular, the series of Fibonacci Lucas Perrin Padovan
Fibonacci's sequence
Repfigits or Keith's numbers [page actually in French, translation in progress].
Douglas Hofstadter's sequences
Sequences whose behavior can be chaotic, described in the book 'Gödel Escher Bach' by Douglas hofstadter.
Java applet for "The Hofstadter Q-sequence".
Somos's sequences
u
n is the quotient of a quadratic form of the coefficients between
u
n-1 et u
n-k+1 by u
n-k
3x+1 problem Syracuse, Collatz or Ulam's sequence ...
Stöhr's
sequences>
u
n is the smallest integer which is not the sum of h preceding distinct elements, at most, of the sequence.
Ulam's sequences
u
n is the smallest integer sum of two preceding terms distinct from the sequence.
A number of magic squares of sums 2
A number of magic squares with coefficients 0, 1, or 2 whose lines and columns (not diagonals) have as sums 2. (A000681 Continuations and A001499 of the encyclopaedia of Sloane)
Sum of same nth powers of the first naturals
In an elementary way, introducing the numbers of Bernoulli. The successive polynomials are calculated using formal integrations.
The Ackerman function
Permutations, derangements, alternating permutations down-up or up-down. Sequences n!, d(n), Euler-Bernoulli.
Number of pairings of the first 2n integers so that the differences of each pair are different.
A060963 This sequence is extended :
1, 1, 5, 29, 145, 957, 8397, 85169, 944221, 11639417, 160699437, 2430145085, 39776366397 ...
On this page (in french), some results about the Perfect Rythmic Tilings (near Skolem or Langford sequences).
(The problem is the search of a maximum clique in a graph)
Restricted Nim and fractal sequences sprague-Grundy functions.
Skolem sequences The number of sequences is 1,0,0,6,10,0,0,504,2636,0,0,455936,3040560 ... or 1, 0, 0, 3, 5, 0, 0, 252, 1318, 0, 0, 227968, 1520280 ... without their symetrics. Skolem triple systel. Steiner triple system. (Page in french language). Java applet for skolem sequences.
Harmonic Numbers The harmonic number H
n=1/1 + 1/2 + 1/3 + ... + 1/n is the sum of the inverses of the n first positive integers. When you reduce H
n the sequences of numerators and denominators are
A001008 (Wolstenholme numbers) and
A002805.
Sequences u(k n) = k u(n), k fixed.
The homothétie of rapport k transform the graph G = {M(n, u(n))} in a subset of G. The sequence
A053644 (a) is in the family. (Page in french language, comprehensible applet).
a) 0,1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,32,32,32,32,32,32,32,
32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,64,64,64,
b) 0,1,2,4,4,8,8,8,8,16,16,16,16,16,16,16,16,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,64,64,64,64,
c) 0,1,1,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,
d) 0,2,2,6,2,2,6,2,2,18,2,2,6,2,2,6,2,2,18,2,2,6,2,2,6,2,2,54,2,2,6,2,2,6,2,2,18,2,2,6,2,2,6,2,2,18,2,2,
6,2,2,6,2,2,54,2,2,6,2,2,6,2,2,18,2,2,6,2,2,6,2,2,18,2,2,6,2,2,6,2,2,162,2,2,6,2,2,6,2,2,18,2,2,6,2,2,
e) 0,-1,-5,-3,-5,2,-15,3,2,-9,-4,0,-15,-3,4,6,1,2,-45,2,-3,9,-4,-3,6,-2,-3,-27,2,0,-12,-3,0,0,3,-1,-45,2,
2,-9,-2,-1,12,1,3,18,4,-5,3,3,2,6,3,2,-135,-4,-3,6,3,-5,-9,4,-3,27,2,-3,-12,-3,-1,-9,-2,-2,18,-2,-5,-6,
Binary trees and total decompositions In french. (Éric Angelini's sequence).