A sequencewhere is a Sheffer sequence, is invertible, and ranges over the real numbers. If is an associated Sheffer sequence, then is called a cross sequence. If , thenis called an Appell cross sequence.An example is the Laguerre polynomial.
A triple of positive integers satisfying is said to be geometric if . In particular, such a triple is geometric if its terms form a geometric sequence with common ratio whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of geometric triples and the set of equivalence classes of harmonic triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .
Let be the th Fibonacci number. Then the sequence is complete, even if one is restricted to subsequences in which no two consecutive terms are both passed over (until the desired total is reached; Brown 1965, Honsberger 1985).
The tangent numbers, also called a zag number, andgiven by(1)where is a Bernoulli number, are numbers that can be defined either in terms of a generating function given as the Maclaurin series of or as the numbers of alternating permutations on , 3, 5, 7, ... symbols (where permutations that are the reverses of one another counted as equivalent). The first few for , 2, ... are 1, 2, 16, 272, 7936, ... (OEIS A000182).For example, the reversal-nonequivalent alternating permutations on and 3 numbers are , and , , respectively.The tangent numbers have the generating function(2)(3)(4)Shanks (1967) defines a generalization of the tangent numbers by(5)where is a Dirichlet L-series, giving the special case(6)The following table gives the first few values of for , 2, ....OEIS1A0001821, 2, 16, 272, 7936, ...2A0004641, 11, 361, 24611, ...3A0001912, 46, 3362, 515086, ...4A0003184, 128, 16384, 4456448, ...5A0003204, 272, 55744, 23750912, ...6A0004116,..
A -automatic set is a set of integers whose base- representations form a regular language, i.e., a language accepted by a finite automaton or state machine. If bases and are incompatible (do not have a common power) and if an -automatic set and -automatic set are both of density 0 over the integers, then it is believed that is finite. However, this problem has not been settled.Some automatic sets, such as the 2-automatic consisting of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (OEIS A048645) have a simple arithmetic expression. However, this is not the case for general -automatic sets.
A sequence of real numbers is equidistributed on an interval if the probability of finding in any subinterval is proportional to the subinterval length. The points of an equidistributed sequence form a dense set on the interval .However, dense sets need not necessarily be equidistributed. For example, , where is the fractional part, is dense in but not equidistributed, as illustrated above for to 5000 (left) and to (right)Hardy and Littlewood (1914) proved that the sequence , of power fractional parts is equidistributed for almost all real numbers (i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive integers, the silver ratio (Finch 2003), and the golden ratio .The top set of above plots show the values of for equal to e, the Euler-Mascheroni constant , the golden ratio , and pi. Similarly, the bottom set of above plots show a histogram of the distribution of for these constants. Note that while most..
A sequence of polynomials , for , 1, 2, ..., where is exactly of degree for all .
There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series(1)into a series with more rapid convergence to the same value to(2)where the forward difference is defined by(3)(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the fact that when is taken in the second of Euler's hypergeometric transformations(4)where is a hypergeometric function, it gives Euler's convergence improvement transformation of the series (Abramowitz and Stegun 1972, p. 555).The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21)...
Define the juggler sequence for a positive integer as the sequence of numbers produced by the iteration(1)where denotes the floor function. For example, the sequence produced starting with the number 77 is 77, 675, 17537, 2322378, 1523, 59436, 243, 3787, 233046, 482, 21, 96, 9, 27, 140, 11, 36, 6, 2, 1.Rather surprisingly, all integers appear to eventually reach 1, a conjecture that holds at least up to (E. W. Weisstein, Jan. 23, 2006). The numbers of steps needed to reach 1 for starting values of , 2, ... are 0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, ... (OEIS A007320), plotted above. The high-water marks for numbers of steps are 0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, ... (OEIS A095908), which occur for starting values of 1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, ... (OEIS A094679).The smallest integers requiring steps to reach 1 for , 2, ... are 1, 2, 4, 16, 7, 5, 3, 9, 33, 19, 81, 25, 353, ... (OEIS A094670)...
The integer sequence defined by the recurrence(1)with the initial conditions , , . This recurrence relation is the same as that for the Padovan sequence but with different initial conditions. The first few terms for , 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers). is the solution of a third-order linear homogeneous recurrence equation having characteristic equation(2)Denoting the roots of this equation by , , and , with the unique real root, the solution is then(3)Here,(4)is the plastic constant , which is also given by the limit(5)The asymptotic behavior of is(6)The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEIS A074788), which occur for terms , 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042,..
A binary plot of an integer sequence is a plot of the binary representations of successive terms where each term is represented as a column of bits with 1s colored black and 0s colored white. The columns are then placed side-by-side to yield an array of colored squares. Several examples are shown above for the positive integers , square numbers , Fibonacci numbers , and binomial coefficients .Binary plots can be extended to rational number sequences by placing the binary representations of numerators on top, and denominators on bottom, as illustrated above for the sequence .Similarly, by using other bases and coloring the base- digits differently, binary plots can be extended to n-ary plots.