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User:Antti Karttunen

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  • For a couple of years, I have been teaching programming, in Assembly, Erlang and Scheme (Racket) programming languages, to people with varying levels of existing skills. (If you know any open position in that field, please leave me a tip! Also for any programming jobs involving discrete mathematics)
  • Previously I have earned my living by various means, anything from washing dishes and demolishing ("maintaining") ships to free-lance journalism and software engineering.
  • I studied in the University of Helsinki mathematics and general & computational linguistics, among other things.
  • Involved with OEIS since 1997 (or was it 1998?). I think A033538 is one of my earliest contributions, but alas, the entries were not yet dated at that time.
  • My mathematical skills are limited to the elementary number theory and combinatorics, with some group theory, as long as it is not too high-brow. I know in principle how generating functions work, but in practice I don't want to operate with them.
  • I don't feel myself really as a mathematician in a classic sense as "somebody who tries to prove unsolved conjectures", but more just as a programmer with some vague sense of aesthetics how to combine various elements from the combinatorics and number theory, activity which sometimes entails interesting further results. Of course I'm not the only person involved with OEIS who has this kind of inclination.
  • Yes, I should also edit other people's sequences. Naturally, I stay away from sequences which I do not understand.
  • Programming languages: The less typing, the more I like them. Also, certain vagueness is for good.

IntSeq-library and other useful packages

The IntSeq Scheme library is located here. (Or will be!)

The Scheme-code for re-creating the A014486/a014486.pdf can now be found here: as a SHAR-archive.

The Scheme-code for all the Catalan bijections I have ever submitted (also some unsubmitted) can be found archived under my github-pages. See for module gatomorf.scm, and CatBijections.pdf for the related draft of my Introductory Survey of Catalan Automorphisms and Bijections paper, never finished.

My favourite sequences

One of my all-time favourites is permutation A122111 from Franklin T. Adams-Watters, especially as it can be formulated as such an elegant recurrence based on prime-shifts. Also Doudna's A005940, which I consider quite an important sequence (see e.g. A278222 for one application), not just some freshman curiosity.

BTW, Matthew Vandermast's formula for A046523, a(n) = A124859(A124859(n)) is very neat!

One interesting array is A114537, Clark Kimberling's "Dispersion of the primes", as it in a kind of way turns the roles of primes and composites totally upside-down, as it is the composites that "originate" the primes there, not the other way.

Some of my best ideas I have borrowed from other people. Now and then I spot an interesting sequence, and start generalizing its theme. For example,

- Wouter Meeussen's A038776, which started the whole "Gatomorphisms" (nowadays officially: "Catalan Automorphisms" or "Catalan Bijections") enterprise, see e.g. table A089840 and its "recursive derivatives". In retrospect, it is curious that the first instance in OEIS (apart from the trivial identity element A001477) is such a highly complex specimen.

- Carl R. White's A179016, which started the "Beanstalk" theme, from which I will slowly churn more variants. (BTW, this topic meets the previous at sequences A218776A218782).

- Katarzyna Matylla's A135141 which I found, thanks to Tony Noe, as he had selected it as the still frame for the OEIS Movie. From the generalization of this come forth the "Entanglement permutations", which many, are not just visually stunning, but have interesting mathematical properties as well. Note that Christopher Eltschka's A071574 is a kind of precedent to A135141, although reflected and using a slightly different starting condition and indexing. In A269847 I compose it with Leroy Quet's A163511 (itself a mirror-image of A005940) to create some more "weirdness".

I have a short to-be-extended draft about these. You can download it here. (And yes, some parts in it are a bit speculative or even childish.)

Of the recent such sequences A285332 is on the verge of whether it is really a permutation or not. Note that I should mention in the future version of the above draft what are the conditions for the pair of complementary sequences that are "entangled" that the result were guaranteed to be a permutation (injective and surjective). At least if one of the components contains cycles, the result cannot be a permutation.

Another kind of entanglement is A283477, which although not a permutation, is still useful "multiplicative encoding" kind of sequence.

Also I like A061773 and similar sequences related to Matula-Goebel numbers: they are concerned with trees, and although they are not planar (oriented), I want to hug them anyway. (Connection with the former kind of trees is given by A127301).

I also like Hugo Pfoertner's shoe lacing sequences: A078698, A078700, A078702, A072503, A079410.

One of the favourites among my own submissions is A051258 ("Fibocyclotomic numbers", numbers formed from cyclotomic polynomials and Fibonacci numbers), one of the few which Neil Sloane has given "nice" keyword. I submitted this in Oct 24 1999 and never viewed it myself anything but a curiosity. However, more than eleven years later, Clark Kimberling has found an interesting application for it. Please see A192233 and A192232.

My favourite graphs

New: Look at Paul Tek's A261892!

New 2: See A276445, which is related to Hofstadter-Conway sequence A004001 and Gray code. Where does the "phase change" at each level y = 2^k + 2^(k-1) ultimately come from?

Katarzyna Matylla's sequence A135141 mentioned above has a very nice graph. Indeed, some other variants, like A237427, look very similar.

Other sequences with elegant or interesting graphs: A161924, A161919, A166166, A129594, A218789 and ratio A218543/A218542.

Also, the ratio of number of odious primes/number of evil primes in each range 2n..2n + 1 A095005/A095006. (Why so much favoritism towards odious primes at first? I can guess some reasons but are they enough?)

Where is the ratio A232742/A232741 converging to? And what about A232745/A232744 ?

Yes, Recaman's sequence, when plotted in special way, A005132 indeed looks quite artistic.

Please see also Index entry for sequences with interesting graphs/plots.

Mathematical speculations and rants

My old home-pages

My old home-pages at i.e. are gone now. For the moment, please search under the archived pages at Internet Archive. I try to move the most important source files and other mathematical material here, time permitting.


  • "On Pascal's Triangle Modulo 2 in Fibonacci Representation", The Fibonacci Quarterly, Vol. 42, 1 (February 2004) pp. 38–46,

Online, Another, a slightly different revision in the Internet Archive.

In this paper I examine and prove in detail how the successive generations of Wolfram's "Rule 90" cellular automata (starting from the single initial seed cell, i.e. Pascal's Triangle computed modulo 2), when interpreted as numbers in the Fibonacci number system (aka Zeckendorf expansion, see A014417), can be surprisingly also computed as products of certain Fibonacci and Lucas numbers. Concerned with the sequence A048757 and other rows of A050609. Specifically, I prove that the following formula holds for all integers n \ge 0, d \ge 0.

\sum_{i=0}^{2n}\,{{2n\choose i}_{\pmod{2}}}\;F_{i+d} = E_{n+d}\prod_{i=0}^\infty\,{L_{2^i}}^{bit_i(n)}

where bit_i(n)\; \overset{\underset{\mathrm{def}}{}}{=} \;{\lfloor{n \over {2^i}}\rfloor\pmod{2}} and En + d stands for Fn + d (the n + d:th Fibonacci number) if n is even, and Ln + d (the n + d:th Lucas number) if n is odd.

  • "Tuhansien lukujonojen aarreaitta" (in Finnish), Helsingin Sanomat, page D1, November 9 2004.

A whole-page article about Neil Sloane's OEIS-database, published in the Science and Nature Section of the leading newspaper of Finland. Online, full text requires paid access.

  • "Series help-mates in 8 and 9 moves" (Problems 2974–2976, together with Olli Heimo), Suomen Tehtäväniekat (Proceedings of the Finnish Chess Problem Society), vol. 60, no. 2/2006, p. 75.

These chess end-game problems of serieshelpmate variety were answers to the challenge left by Richard Stanley in his Queue problems revisited (Please see page 8 of the PS/PDF-file on his site). These problems involve "zig-zag posets", so the number of solutions for n-move problem is given by the nth term of A000111.

  • Several articles published 2005 in Prosessori concerning functional programming languages, libraries and the character sets.


Table of the most coherent themes for which I have contributed sequences, (other than Catalan Automorphisms, that will be explained later)

Number of times certain "simple" recursive programs call themselves (written for example in Forth or Lisp) A033538A033539
Sequences found from or inspired by HAKMEM A036213A036214, A048707A048708
Periodic vertical binary vectors and "sloping binary representation" for the Fibonacci numbers and for the powers of 3. A036284, A037096, A037093, A037095
Patterns of binomial coefficients computed modulo 2, reduced residue sets of integers, or cyclotomic polynomials reinterpreted as binary strings, Zeckendorf expansion, or in similar representation. Note that in many cases there is an interesting interplay between two mathematical domains. (E.g. the divisibility rules of Fibonacci numbers and how the Cyclotomic Polynomials are constructed). A038183, A048757, A054432, A054433, A063683, A051258, A055094
Infinite sequences of finitary permutations, listed in some order, their inverses, permuted by Foata transform, etc. Still to do: the corresponding composition ("multiplication") tables. A055089 & A060117, A056019 & A060125, A065181A065183
Sequences related to bell-ringing (campanology) and similar ideas about Hamiltonian circuits through symmetric groups (much more to explore here!) A057112, A060135 & A060112.
Number of permutations and derangements up to rotations A061417 & A064636.
Stern-Brocot tree related permutations and other sequences. The first one "contains" (i.e. are easily mapped to) a subset of elements of Thompson's group F, although I didn't realize that at the submission time. (Note that actually Thompson's groups have been defined to act on the infinite binary tree of dyadic rationals, but in any case, the underlying binary tree is of the same shape as SB-tree, so the same rotations can be applied in either case). A065625, A065658, A054424 & A054425, A054427, A065936 & A065937
Kepler's tree of harmonic fractions A086592
Number of certain siteswap configurations (juggling diagrams) A065177, A084509, A084519, A084529
Infinite sequences of juggling siteswaps for three balls (somebody could compute the 2- and 4-ball analogues) A084501, A084511, A084521, A084452, A084458
Counts of different varieties of primes in [2n,2n+1] A095005A095024, A095052A095069, A095731, A095741. Further exploring in A095759
Sequences related to Legendre- & Jacobi-symbols of primes/odd numbers, and their partials sums A080114, A095102, A095100, A112070, A166092, A166040, A165601, A165603, etc.
GF(2)[X]-polynomials Many of the sequences in the index for GF(2)[X]-polynomials. See especially A091202A091205 & A106442A106446.
"Congruent products" between domains N and GF(2)[X] Most of the sequences in the index for congruent products between domains N and GF(2)[X]

and also in the index for congruent products under XOR. This started from "fibbinary-like" sequences A048715, A048716 & A048718, with further inspiration from Paul D. Hanna. More exploration in A115857 & A115872. The name of seq. A115823 is incorrect, should probably be 21 instead of 23, have to check.

Sequences related to Combinatorial Games encoded as binary strings A106485A106487, A126000, A126011
Number of linear (and alternating linear) extensions of the divisor lattice of n (together with Mitch Harris). A114717, A119842
Hilbert's Hamiltonian walks in the infinite square grid A163334 & A163357, A165465 & A165467
Miscellaneous chess related sequences A062104, A065188, A065256, A062103
Miscellaneous number theoretic sequences A055095, A055096, A059871
Permutation of natural numbers involving XOR (together with Paul D. Hanna). Note that this is (probably) the only permutation of N based on a greedy algorithm, which I have contributed to, although such permutations are quite popular in OEIS. (And yes, the idea was Paul's.) For some reason, I'm not into them. A116626
Miscellaneous permutations of natural numbers. (Apart from signature-permutations of Catalan automorphisms). There are a lot's of them. For example: A129594 has an intriguing graph.
Multiplicative permutations. In January 2014 I got interested about the multiplicative permutations (that is, really multiplicative in Z, unlike A091202-A091205 and their kin). The most interesting variants are ones that act also as some kind of automorphisms of nonoriented rooted trees, as I had always thought that there are essentially none apart from the identity. But in contrast to my dear plane rooted trees in Catalania, where the automorphisms "begin from the root", it seems that automorphisms/bijections for non-oriented trees are easiest to determine to act on their terminal branches. See here
A couple of sequences related to calendars or numbering systems A098378, A098476
A sequence related to linguistics (see User:Antti Karttunen/Etsivät etsivät etsivät) A213705

In the kettle



I uploaded (as a test) my Python checking script for OEIS-entries. It is probably in wrong place: (How this should be referenced?)

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