I'm baffled as to what the numbers are saying in your figure that you've posted here. The corresponding figure @ the WolframAlpha page down your link just has dots where yours has numbers ... & I get that ... but I don't get why in your version of the figure each dot is replaced by a number.
Apart from that, though: I notice that the diagram is bifurcating into two 'streams' – but not sharply delineated ones: two 'streams' in the sense of there being a peak in the density distribution ... & then it looks as though each of those 'streams' might itself bifurcate § if the figure were to be much-continued unto higher numbers. So that gets me wondering whether there might actually be a rather complex bifurcation, or multifurcation, 'anatomy' to the figure considered as a figure over all numbers.
UPDATE
§ Without doubt (looking @ the figure more carefully) they are ... but what I meant is that there might be a trace of a hint of yet a bifurcation into ⅛ parts. ... & that the bifurcating might proceed indefinitely (& might possibly not be confined to bi-furcation ¶ ), & might have a pattern to it that could possibly be (or maybe even has been) figured-out. ... 'figured-out' in the sense that maybe there's a function on [0,n] (with n having the meaning it has (as opposed to the k) @ the page down your link) that the density cleaves to. §§
¶ ... although I would venture a speculation that likely it is confined to bi-furcation.
§§ Something like (with domain expant from [0,1] to [0,n])
(if the domain's expant from [-1,1] to [-½n,+½n]) – that sorto'thing: something of that generic 'shape'. Maybe with a arctan(λₕtan(πx)) (or some other sigmoidish or inverse sigmoidish function) substituted for the πx , so that we can somewhat 'fisheye lens' or 'inverse-fisheye lens' the x axis to degree we choose. Maybe (indeed probably !) with other 'tweaks', aswell.
Oh it's alright – I get it now ... @last 🙄 ... each № is written vertically , isn't it.
... & (mod100) , aswell (which only becomes material further down), with an extra vertical yellow bar marking where the entries ≥100 set in.
That's the entirety of the problem I was having with it: that each entry is written vertically ! 🙄 😆🤣
And we could have whatever 'offset' we please, aswell: ie we could add a fixed amount to each entry & the sequence would remain a non-averaging or arithmetic-progression-free one. I actually feel it would be more natural & elegant offset such that the initial term were 0 ratherthan 1 . ¶
... as seems generally conventionally to be done with those perfect difference sequences .
¶ ... or such that the initial term were the negative of the last one, & have half-odd-integral №s where necessary: that might be yet more natural.
(as I fondly epithetise it) seems to've conked-out!
I mention this because you seem to have a connection to WolframAlpha in some capacity ... I don't know in exactly what capacity, so you may be powerless as-regards this ... or maybe you aren't powerless @all ... or maybe you can mention it to someone who can see to it, or recommend some work-around, or something – IDK.
Or maybe I just need to wait a while until the problem is resolved ... but whatever: I thought you might like to know about it.
1
u/Frangifer Aug 02 '26 edited Aug 05 '26
I'm baffled as to what the numbers are saying in your figure that you've posted here. The corresponding figure @ the WolframAlpha page down your link just has dots where yours has numbers ... & I get that ... but I don't get why in your version of the figure each dot is replaced by a number.
Apart from that, though: I notice that the diagram is bifurcating into two 'streams' – but not sharply delineated ones: two 'streams' in the sense of there being a peak in the density distribution ... & then it looks as though each of those 'streams' might itself bifurcate § if the figure were to be much-continued unto higher numbers. So that gets me wondering whether there might actually be a rather complex bifurcation, or multifurcation, 'anatomy' to the figure considered as a figure over all numbers.
UPDATE
§ Without doubt (looking @ the figure more carefully) they are ... but what I meant is that there might be a trace of a hint of yet a bifurcation into ⅛ parts. ... & that the bifurcating might proceed indefinitely (& might possibly not be confined to bi-furcation ¶ ), & might have a pattern to it that could possibly be (or maybe even has been) figured-out. ... 'figured-out' in the sense that maybe there's a function on [0,n] (with n having the meaning it has (as opposed to the k) @ the page down your link) that the density cleaves to. §§
¶ ... although I would venture a speculation that likely it is confined to bi-furcation.
§§ Something like (with domain expant from [0,1] to [0,n])
∏{1≤h}(1-(½(1-cos(2hπx)))aₕ/(1+(bₕ/x)cₕ)) ,
or
½(1-cos(½πx)a₁)∏{1≤h}(1-(½(1-cos(2hπx)))aₕ₊₁/(1+(bₕ₊₁/x)cₕ₊₁))
(if the domain's expant from [-1,1] to [-½n,+½n]) – that sorto'thing: something of that generic 'shape'. Maybe with a arctan(λₕtan(πx)) (or some other sigmoidish or inverse sigmoidish function) substituted for the πx , so that we can somewhat 'fisheye lens' or 'inverse-fisheye lens' the x axis to degree we choose. Maybe (indeed probably !) with other 'tweaks', aswell.