User:Nathan L. Skirrow

helpful links

sequences I authored, edit history, currently-open drafts search results for my name (containing notes upon other sequences)

Me

second-year maths undergraduate

I have a Github account, and a LifeWiki account (with somewhat more of a synopsis than this), and have made many more mainspace contributions there than here (many of which are from using my Life simulator to identify interesting and notable patterns in Catagolue, but a few technical articles as well).

my mathematical interests are largely combinatorial, most of my knowledge thereto came from this very site

highly autistic and 🏳️‍⚧️

userspace pages

approximately in order of advancedness

uninteresting numberbashery

• /useful polynomiality: brief covering of cubic formula followed by solving trinomials with hypergeometrics! (written before finding M. Lawrence Glasser's paper, which largely supersedes it)
• /A156744 and square hyperpyramids: small connection between sums of reciprocals of a set of figurate numbers and a sequence about fitting polynomials

interesting combinatoriality

• /bitwise permutations: on some algorithms for computing A060351 and A368070 and an upper bound
• rook-theoretic interpretations of Stirling numbers!

• /Stirling factoradics: on cycle numbers (vertically-nonattacking rook configurations in triangles)
• /rook configurations: on subset numbers (bidirectionally-nonattacking)

• /A058349: incomplete research project, on a bijection between disparate tree structures on labelled sets

• /some number triangle matrix products: thing that sprouted from my knowledge of certain kinds of g.f. therefrom

my sequences

My first large endeavour in Python (in which my understanding became functionally complete) was my tablebase program, which renders and simulates state transition diagrams in 3D, and led to the creation of A357723 and A357740 (enumerating king positions under symmetry, that arise in tablebases), and A358339 (enumerating positions with each distance to mate in a king and rook vs. king (KRvK) endgame). After learning about the Pólya enumeration theorem, I made A361870 (counting 2-colorings of hypercube arrays under their symmetry groups).

Ideas for future sequences

I have several, but unfortunately the draft approval process very often takes many weeks (or even months) after a sequence is complete, so most of the time I am waiting.

• c-colourings of hexagonal arrays of edge length n+1 (equivalently, number of transitions in c-state INT rules in range-n hexagonal neighbourhoods), (c^3*n*(n+1)+1+3*c^{(3*n+2)*(n+1)/2}+3*c^{(3*n2+4*n+2)//2}+c^{3*n*(n+1)/2+1}+2*cⁿ²⁺ⁿ⁺¹+2*c^n*(n+1)/2+1)/12, probably the sequence will be for c=2 like A054247.
• In One-dimensional cellular automaton/Wolfram rule, I noted that for n>=1, there are (2²ⁿ+2^{(2n+2(n+1)//2)/2}+2^2n-1+n%2)/4 nonequivalent Wolfram rules on a width-n neighbourhood under action of left/right reflection and black/white reversal (where the second term's exponent is A005418(n+1)=A361870(1,n)). (For n=0, probably there should be considered to be 1, from the empty set to itself, or perhaps it should be forgone.) Note that as an enumeration of equivalence classes of boolean functions under four actions (from compositions of the two self-inverse functions: bitwise reversal and NOT of their truth tables (rule integers), equivalently complementing the colours and inverting the axial positions within a hypercube; and reversal of binary representations of bits' indexes (reversing the order of precedence of the axes), it is Θ(2²ⁿ/4), whereas A000616 has a n!*2ⁿ actions so is Θ(2²ⁿ/(n!*2ⁿ)).
• In a similar manner to A361870, I found some equations for 2-colorings of n-dimensional orthoplexes inscribed within edge-length-k hypercubes (equivalently, INT transitions in higher-range von Neumann neighbourhoods in higher-dimensional cellular automata), I would like to make a similar array by antidiagonals for them eventually. As with A361870, a(1,k)=A005418(k+1)=(2^k+2^⌈k/2⌉)/2, however for a(2,k), there is a nice form (in terms of floordivs) over the vertex- and cell-centred cases (but seemingly not both together),

• ( 2^(k2+1)/2+2*2^(k+1)2/4+2*2^{((k+1)//2)2-(k+1)//4}+2^(k2+3)/4+2*2^(k2+7)/8
  .if k≡1 (mod 2) else
  . 2^k*(k+2)/2+2*2^{(k+2)*(k+1)//4}+3*2^k*(k+2)/4+2*2^k*(k+2)/8)/8

• and for a(3,k) (octahedra),

• ( 2^k*(k2+5)/6+3*2^k*(k2+11)/12+3*2^{(k3+3*k2+5*k+3)/12}+6*2^{(2*k3+3*k2+10*k+9)/24}+6*2^{(k3+8*k+6-k%4*3)/12}+2^{(k3+5*k+6)/12}+8*2^{(k3+9*k+8*((k+1)%3-1))/18}+6*2^k*(k2+23)/24+6*2^{(k3+11*k+12)/24}+8*2^{(k3+9*k+18+8*((k+1)%3-1))/36}
  .if k≡1 (mod 2) else
  . 2^{k*(k3+6*k+8)/6}+13*2^{k*(k2+6*k+8)/12}+6*2^{(2*k3+15*k2+28*k+k%4*6)/24}+8*2^{(k3+6*k2+12*k+k%3*8)/18}+12*2^{k*(k2+6*k+8)/24}+8*2^{((k3+6*k2+12*k+k%3*8)/36})/48

Minor things to perhaps note

• The g.f. for A000975 is provided in both factorised and expanded form, however its e.g.f. (4*e^2*x-3*e^x-e^-x)/6 factorises to (1-e^-x)*(4*e^2*x+e^x+1)/6.
• If Vaclav Kotesovec's conjecture for A225552 is correct (which is strongly suggested by the fact that A358339(n,2*A225552(n)) seems to become constant for odd n>=15, I imagine it does for even n>=26 but will need to run my program for longer to check),

• the g.f. is x³*(x²⁸-x²⁷-x²²+x²¹-x¹⁵+x¹⁴-x¹³+x¹²+x¹⁰-x⁸-x⁷-x⁶+2*x⁵+2*x⁴+2*x³+3*x²+4*x+3)/((1-x)²*(1+x)*(1-x+x²)*(1+x+x²)) (the denominator expands to 1-x-x⁶+x⁷, making it linear-recurrent with signature (1,0,0,0,0,1,-1), the one shared by all with 6-periodic first differences)
• the e.g.f. is (7*(x-1)*e^x-e^-x-x³+((x⁶-(x⁹+(x¹¹-11!/24!*x²⁴)/110)/504)/40-sqrt(3)*(3*e^x/2+e^-x/2)*sin(sqrt(3)*x/2)+6*cos(sqrt(3)*x/2)*cosh(x/2))/6)/3+3-x² (however maybe I ought not to put that :-)

things to note about sequences

• Michael Somos's document, Indexing Functions for Triangular or Rectangular Arrays (and many others!), in Python integer arithmetic operations, and their g.f.s in terms of theta functions, in a LifeWiki page
• Colin Barker's conjecture on Jan 05 2016 about the closed form for A266725 (Total number of OFF (white) cells after n iterations of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.) is true (trivially from inspection, though maybe he assumed it would be more chaotic or fractal-like).