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Topic: Integer sequences

	Integer sequence - Wikipedia, the free encyclopedia
		An integer sequence is a definable sequence, if there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences.
		The set of computable integer sequences and definable integer sequences are both countable, with the computable sequences a proper subset of the definable sequences.
		The set of all integer sequences is uncountable; thus, almost all integer sequences are uncomputable and cannot be defined.
	en.wikipedia.org /wiki/Integer_sequence (207 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Sequence M0255, whose first few terms are 0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4..., turns out to be the minimum number of multiplications needed to compute an nth power.
		This is sequence M0208, and from the rule for sorting the sequences you may be able to guess its identity: the all-2s sequence, 2, 2, 2, 2....
		One example is the sequence in which a(n) is the integer nearest to the base-10 logarithm of n; it includes 285 2s followed by 2,846 3s.
	www.cecm.sfu.ca /organics/papers/bailey/paper/html/compsci96-01.html (3610 words)

	Shaft Sequences
		This page contains links to pages for a variety of integer sequences that have been converted to "shaft arithmetic" for use in making threading and treadling sequences (the use of "shaft" is arbitrary; the concept applies equally well to treadles).
		The sequence of positive integers (natural numbers) is the most basic of all integer sequences.
		The Fibonacci sequence is arguably the most studied of all integer sequences.
	www.cs.arizona.edu /patterns/sequences/sequences.html (867 words)

	Encyclopedia of Combinatorial Structures (Introduction)
		Sloane's Encyclopedia of Integer Sequences with an emphasis on sequences that arise in the context of decomposable combinatorial structures.
		A sequence of integers: the (n+1)th term of this sequence is the number of objects of size n defined by the specification.
		When the sequence (f(n)) is in Sloane's Encyclopedia of Integer Sequences, the references contain "EIS nb" with nb the sequence number in the EIS.
	algo.inria.fr /encyclopedia (366 words)

	Special Integer Sequences (Site not responding. Last check: 2007-10-13)
		A reasonable rule for generating this sequence is that the integer n appears exactly n times, so the next five terms of the sequence would all be 5, the following six terms would all be 6, and so on.
		For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list.
		The terms of this sequence differ from the terms of the given sequence by 1, 2, 3, 4, 5,....
	cse.unl.edu /~aendo/SpecialIntegerSequences.html (490 words)

	The CTK Exchange Forums
		As a = SQRT(8q^2 + 1), this sequence can be used to generate the n, n + 1, , an - (a + 1) / 2 sequences, the sum of each of which will be a square, q^2 (2n — 1)^2.
		On the other hand you did make a significant contribution to the encyclopedia regarding these sequences and sum of consecutive numbers equaling a square or cube and the fact that you specified the recursive relation incorrectly might be correctable.
		These sequences are a generalized form of that for square triangular numbers.
	www.cut-the-knot.com /htdocs/dcforum/DCForumID4/602.shtml (737 words)

	Integer Sequences Related To PI
		Integer Sequences Related To PI Integer Sequences Related To PI For any given integers s[0] and s[1] we can construct the infinite sequence s[n], n=0,1,2,...
		The ratio of the 9th terms of these sequences is 61445088 3 --------- = 0.356194984...
		By the way, since (3) shows that all the sequences are easily related, it's enough to consider just the ratio of two sequences.
	www.mathpages.com /home/kmath381.htm (612 words)

	Info on Degree Sequences (Site not responding. Last check: 2007-10-13)
		--> A degree sequence of an undirected graph graph is a monotonically non-increasing sequence of the degrees of its vertices.
		A007721 in Neil J. Sloane's database of integer sequences.
		The number of degree sequences of biconnected graphs for n = 3,4,...,15, is 1, 3, 9, 34, 125, 473, 1779, 6732, 25492, 96927, 369463, 1412700.
	www.theory.csc.uvic.ca /~cos/inf/nump/DegreeSequences.html (251 words)

	Welcome to the On-Line Encyclopedia of Integer Sequences
		If your sequence isn't in the database, and if it is interesting, please submit it using the web page for contributing a new sequence or comment.
		Most of the sequences are arranged in the database in lexicographic order of absolute values, indexed by the position of the first term that is greater than 1 in absolute value.
		Sequences that contain only 0's, 1's and -1's are in strict lexicographic order by absolute value at the beginning of the table.
	www.research.att.com /~njas/sequences/Seis.html (1691 words)

	Douglas Hofstadter's sequences. Chaotic sequences.Unsolved problems.
		The first of these sequences is from the book 'Gödel Escher Bach' by Douglas Hofstadter.
		Like the Fibonacci and Lucas sequences, each term is the sum of two preceding terms, but not the two last terms:
		The sequence has a moderately erratic behavior, perhaps chaotic, but it is not certain, the graphs let think of a certain form of regularity.
	perso.wanadoo.fr /jean-paul.davalan/mots/suites/hof/index-en.html (331 words)

	Integer Sequences (Site not responding. Last check: 2007-10-13)
		Looking for many of the sequences I had been generating in that Encyclopedia, I found that several were not yet entered there.
		This sequence is n(n+3)/2, at A000096, where it is given starting at zero.
		Clearly, there is little new in these geometric integers, but other sequences can be derived by combinations of functions from the two groups.
	www.comp.utas.edu.au /users/nholmes/sqncs (520 words)

	Exactly realizable sequences
		It is easy to show that a sequence is exactly realizable if and only if its image under ORBIT is also a sequence of non-negative integers (see Puri's thesis [1] for the details).
		All the sequences are expected to have realizing maps - the inclusion of a map means that we know of a map that is natural in some sense (for example, has a finite description or is algebraic).
		Sequences marked with a question mark in the first column are not known to be exactly realizable at all: they just seem to satisfy the congruence for the first twenty or so terms.
	www.mth.uea.ac.uk /%7Eh720/research/files/integersequences.html (529 words)

	Organizing Thoughts into Sequences, Hierarchies, and Networks (Site not responding. Last check: 2007-10-13)
		The results of this survey indicate that an important theoretical distinction should be made between the three external structurings (sequence, hierarchy, network), and a fourth kind of structuring, the internal structure of records (Section 6).
		It first presumes a hierarchy of classes, and then establishes a sequence of priority, one that happens to favor the method defined in an outer class over the method defined in an inner class when the two have the same name.
		Sequences, hierarchies, and networks are three external structurings that together account for visualizations in terms of chronicles, evolutions, catalogs, atlases, canons, and tours.
	www.ms.lt /ms/projects/structurekinds/paper052499.html (4742 words)

	The South End Newspaper - Mathematician reaches 100k milestone for online integer archive - NATION/WORLD - News (Site not responding. Last check: 2007-10-13)
		There are sequences in the OEIS that relate to physics — such as the centered cube numbers that relate to shells of atoms — biology and even music (such as 2, 2, 4, 4, 2, 6, 6, 2, 8, 8, 16, which is in the lyrics of an Argentine children’s song).
		Sloane started rounding up integer sequences in the 1960s, entering them on punch cards, when he was working on neural networks as a graduate student at Cornell University.
		Although Sloane acknowledges that all the “core” sequences — such as the prime numbers, Catalan numbers, and the Fibonacci sequence — are already in the database, he believes the OEIS has an infinite potential for expansion.
	www.southend.wayne.edu /modules/news/article.php?storyid=553 (1221 words)

	SIAM Review of ``An Encyclopedia of Integer Sequences'' by N. J. A. Sloane & Simon Plouffe
		The idea is that a researcher who encounters a sequence in her or his work, and wishes to quickly find out what is known about the sequence (does it have a name, for example, such as ``the Euler numbers'' or ``the Stirling numbers of the first kind''?), can look it up here.
		Some of the heuristics discussed in chapters 1, 2, and 3 (before the table of sequences proper begins) give useful hints as to what to do when the computer programs don't work; they also give a nice conceptual model of the inner workings of the programs.
		Further, about 25% of the sequences in the book are obtained from a rational generating function or elementary manipulation thereof (reversion, undoing a logarithmic differentiation, etc.).
	www.cecm.sfu.ca /~jborwein/sloane/sloane.html (630 words)

	Number of Primes arising as Sum of a Factorization (Site not responding. Last check: 2007-10-13)
		The sequence had the attributes "more" and "hard", and I thougth "more" is right but "hard" is not.
		It is the sequence of first occurances of n in the original sequence.
		Here are the first 30 values of the sequence of first occurances, from 0 to 29: 8, 2, 6, 90, 30, 390, 690, 420, 210, 4290, 3990, 8778, 2310, 3570, 4830, 11550, 38850, 84630, 66990, 79170, 39270, 30030, 51870, 46410, 43890, 111930, 163020, 221340, 419430, 131670.
	www.people.freenet.de /nQueens/SumOfFactorPrimes.html (539 words)

	Classroom Activities Using Number Patterns
		Add the number 8 to the sequence, and ask the students to write a guess for the next number (13).
		Ask the students to explain the rule that generates the sequence (doubling the previous number.) Tell them these are the binary numbers.
		If they haven't seen this sequence before, draw a triangular array of dots on the board with 1 dot in the first row, 2 dots in the second row, 3 dots in the third row, and so on.
	www.dpgraph.com /janine/mathpage/patterns.html (2107 words)

	Numbers and Functions as Continued Fractions - Numericana
		Continued fractions for which the sequence of partial quotients is ultimately periodic are called periodic continued fractions and they correspond to quadratic irrationals [also called algebraic numbers of degree 2, these are irrational roots of polynomials of degree 2 with integral coefficients].
		A003285 in the Online Encyclopedia of Integer Sequences for the lengths of the periods of the square roots of the successive integers (by convention, a finite CFE has zero "period").
		Keep in mind that all integers involved in the expansion must be positive integers (with the possible exception of the leading one) so that a lot of case splitting is to be expected.
	home.att.net /~numericana/answer/fractions.htm (3614 words)

	My Project
		The original paper analyzing this sequence was written by Blum in an article in 1974.
		We were looking at the integer sequence a_n which enumerates the number of square permutations of fixed length n (mathematical background, including the definition of square permutations, can be found in the paper below).
		In 1974, when this sequence was analyzed, a difficult proof with advanced combinatorics was given to prove that the sequence satisfied the recurrence a_(2n) (2n+1) = a_(2n+1).
	www.tjhsst.edu /%7Eyzhang/techlab/proj4/index_rec.html (685 words)

	SS > NF reviews > N. J. A. Sloane (Site not responding. Last check: 2007-10-13)
		Most of these sequences are of genuine mathematical interest (Fibonacci sequence, Catalan numbers, polynomial expansions, primes, continued fractions, to name but a few) or physical interest (magnetisation and susceptibility for square and cubic lattices, atomic weights) -- a few are just for fun (number of letters if written in Roman numerals).
		There is even a table, Figure M4822, of 'puzzle sequences' that aren't in the main body because they don't satisfy the rules for inclusion: (mostly) infinite integer sequences.
		The long list of sequences is leavened with the occasional diagram showing a geometrical interpretation of a particular sequence.
	www-users.cs.york.ac.uk /susan/bib/nf/s/sloane.htm (284 words)

	Derived Sequences - Cohen, Iannucci (ResearchIndex) (Site not responding. Last check: 2007-10-13)
		Abstract: We de ne a multiplicative arithmetic function D by assigning D(p, when p is a prime and a is a positive integer, and, for n 1, we set D (n)) when k 1.
		We term fD k=0 the derived sequence of n.
		We show that all derived sequences of n < 1:5 10 are bounded, and that the density of those n 2 N with bounded derived sequences exceeds 0.996, but we conjecture nonetheless the existence of unbounded sequences.
	citeseer.csail.mit.edu /589078.html (255 words)

	[No title]
		Subject: Re: integer sequences From: matrng@vaxa.hofstra.edu Date: 19 Jul 94 11:38:48 EST Newsgroups: sci.math I see that no one has responded to my query so far, but no matter, because I finally located the information on getting integer sequences.
		The reply will report all sequences found in the Encyclopedia (up to a limit of 7) that match: your sequence, your sequence with 1 subtracted from each term, your sequence with 1 added to each term.
		This will tell you if the sequence is in the table : Encyclopedia of integer Sequences, actually the number of sequences is 6222.
	www.math.niu.edu /~rusin/known-math/94/number.seq (583 words)

	Number of Irreducible Polynomials over GF(4) of Given Trace and Subtrace (Site not responding. Last check: 2007-10-13)
		A074031 in Neil J. Sloane's database of integer sequences.
		A074032 in Neil J. Sloane's database of integer sequences.
		A074033 in Neil J. Sloane's database of integer sequences.
	www.theory.csc.uvic.ca /%7Ecos/inf/trs/poly/Fq/poly_tr_subtr_F4.html (182 words)

	Amazon.com: Books: The Encyclopedia of Integer Sequences (Site not responding. Last check: 2007-10-13)
		The sequences arearranged in numerical order, and for each one a brief description and a reference is given.
		The number of sequences cataloged here is more than double the tally of the previous incarnation....If libraries shelve this book in the reference section, they should consider aquiring a second copy for circulation.
		There are twice as many sequences as there were in Sloanes Handbook and those who have the Handbook will want The Encyclopedia....Many people who have searched in vain for some of the sequences missing from the Handbook will be quick to get copies of this new and expanded version to track down these missing sequences.
	www.amazon.com /exec/obidos/tg/detail/-/0125586302?v=glance (1635 words)

	INTEGER (Site not responding. Last check: 2007-10-13)
		Almost all mathematicians have read, at one time or another, the words of Leopold Kronecker: "God made the positive integers; all else is the work of man." Almost everyone, whether a mathematician or not, can agree that there is something very, very basic about the numbers 1, 2, 3, 4, 5,.
		Other sequences, less baffling, exhibit patterns - or absense of patterns - whose appeal shines beyond whatever applications these sequence might find outside mathematics.
		By returning to my home page, you can access information about kinds of sequences and arrays (sequences of sequences) that I find especially interesting.
	faculty.evansville.edu /ck6/integer (149 words)

	Math Magic
		The sequence of values of g(n): 4, 6, 4, 8, 4, 9, 4, 6, 4, 11, 4, 11, 4, 6, 4, 12, 4, 13, 4, 6, 4, 13, 4, 8, 4, 6, 4, 14, 4, 15, 4, 6, 4, 8, 4,.
		The sequence of values of G(n): 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13,.
		The sequence of values of h(n): 4, 5, 4, 4, 4, 3, 4, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,.
	www.stetson.edu /~efriedma/mathmagic/1298.html (1250 words)

	Integer Sequence Combinations (Site not responding. Last check: 2007-10-13)
		The nature of the code given in the following examples is explained in the index document.
		Integer sequences which are calculated directly from their indexes can have another calculation applied to them.
		Functions to be used as a basis for the secondary functions defined below are given with explanation and examples in the index file, so their definitions are simply listed in the following.
	www.comp.utas.edu.au /users/nholmes/sqncs/cmbntns.htm (188 words)

	Texts and tools for the Online Encylopedia of Integer Sequences
		Almost 10,000 sequences (May 2003) are not marked as dead, dupe, full, hard, obsc, unkn, or more, where the third line of terms is empty and no term is longer than 30 digits.
		The special case of a sequence without signed terms marked as sign but not nonn is only reported, because the signed terms may start later, see A057752 resp.
		Various Shell sort OEIS sequences are tested: Shell's A003462, Knuth's A033622, and A055876, A055875, A036562, A036564, A036569.
	www.xyzzy.claranet.de /eis.htm (890 words)

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