- mathfun@mailman.xmission.com - mailman.xmission.com

[math-fun] Pick any 4 successive terms a, b, c, d and have a/b = c/d.
by Éric Angelini 22 Oct '19

22 Oct '19

Hello Math-Fun, I've sent this 5 days ago to SeqFans but it seems that it never reached the list. I try here, Best, É. ---------------------------------------------------- Hello (SeqFans) Math-Fun, I was looking for a seq S (of distinct integers) where one could pick any 4 successive terms a, b, c, d and have a/b = c/d. I quickly found https://oeis.org/A038754 [Original definition of the seq is between brackets] [a(2n) = 3^n, a(2n+1) = 2*3^n]. S = 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486,... The above seq starts with 1,2,3. The start 1,2,4 produces the powers of 2. The start 1,2,5 produces https://oeis.org/A026383 [a(n) = 5a(n-2), starting 1,2]. The start 1,2,6, produces https://oeis.org/A026549 [Ratios of successive terms are 2,3,2,3,2,3,2,3...] The start 1,2,7 produces https://oeis.org/A123752 [a(n) = 7*a(n-2), a(0) = 1, a(1) = 2]. The start 1,2,8 produces https://oeis.org/A098232 [Largest power of 2 <= 3^n]. The start 1,2,9 produces https://oeis.org/A083423 [a(n) = (5*3^n + (-3)^n)/6]. The start 1,2,10 produces https://oeis.org/A004643 [Powers of 2 written in base 4]. etc. (I love the rainbow of definitions!) The start 1,3,2 produces https://oeis.org/A164073 [a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3]. The start 1,3,4 produces https://oeis.org/A084221 [a(n+2) = 4*a(n), with a(0)=1, a(1)=3]. The start 1,3,5 produces https://oeis.org/A056487 [a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd]. ....(!?) and produces also https://oeis.org/A111386 [a(1) = 1, a(2) = 3; for n >= 3, take a(n) to be the smallest odd number not occurring earlier such that a(n-1) divides the concatenation a(n-2)a(n)]. The start 1,3,6 produces https://oeis.org/A026532 [Ratios of successive terms are 3,2,3,2,3,2,3,2...] The start 1,3,7 produces 1,3,7,21,49,147,343,1029,... _not in the OEIS_. The start 1,3,8 produces https://oeis.org/A096886 [Expansion of (1+3*x)/(1-8*x^2)]. The start 1,3,9 produces the powers of 3 The start 1,3,10 produces https://oeis.org/A004663 [Powers of 3 written in base 9]. etc. The start 1,4,2 produces https://oeis.org/A143095 [(1, 2, 4, 8,...) interleaved with (4, 8, 16, 32,...)]. But we have repeated terms there. The start 1,4,3 produces https://oeis.org/A166552 [a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4] The start 1,4,5 produces https://oeis.org/A133632 [a(1)=1, a(n)=(p-1)*a(n-1), if n is even, else a(n)=p*a(n-2), where p=5]. The start 1,4,6 produces https://oeis.org/A164532 [a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4]. The start 1,4,7 produces 1,4,7,28,49,196,343,1372,... _nt in the OEIS_ The start 1,4,8 produces https://oeis.org/A094015 [Expansion of (1+4x)/(1-8x^2)]. The start 1,4,9 produces https://oeis.org/A133125 [A133080 * A000244]. The start 1,4,10 produces https://oeis.org/A136859 [Numbers n such that n and the square of n use only the digits 0, 1, 4 and 6]. etc. (another rainbow!) The start 1,5,2 produces 1,5,2,10,4,20,8,40,16,80,32,160,... _not in the OEIS_. The start 1,5,3 produces https://oeis.org/A166465 [a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 5]. The start 1,5,4 produces 1,5,4,20,16,80,64,320,256,... _not in the OEIS_. The start 1,5,6 produces https://oeis.org/A166023 [a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 5] The start 1,5,7 produces 1,5,7,35,49,245,343,... _not in the OEIS_. The start 1,5,8 produces 1,5,8,40,64,320,512,... _not in the OEIS_. The start 1,5,9 produces 1,5,9,45,81,405,729,... _not in the OEIS_. The start 1,5,10 produces https://oeis.org/A268100 [a(n) = 2^((n-1) mod 2)*5*10^floor((n-1)/2)] etc. The start 1,6,2 produces https://oeis.org/A163864 [a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6]. The start 1,6,3 produces https://oeis.org/A166450 [a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 6]. The start 1,6,4 produces https://oeis.org/A081631 [2*2^n-(-2)^n]. The start 1,6,5 produces 1,6,5,30,25,150,125,750,... _not in the OEIS_. The start 1,6,7 produces 1,6,7,42,49,... _not in the OEIS_. The start 1,6,8 produces https://oeis.org/A164640 [a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 6]. The start 1,6,9 produces 1,6,9,54,81,... _not in the OEIS_. The start 1,6,10 produces 1,6,10,60,100,600,1000,6000,10000 _not in the OEIS_. etc. The start 1,7,2 produces 1,7,2,14,4,28,8,56,16,112,32,224,... _not in the OEIS_. The start 1,7,3 produces https://oeis.org/A166481 [a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 7]. The start 1,7,4 produces 1,7,4,28,16,96,64,... _not in the OEIS_. The start 1,7,5 produces 1,7,5,35,25,175,... _not in the OEIS_. The start 1,7,6 produces 1,7,6,42,36,252,... _not in the OEIS_. The start 1,7,8 produces 1,7,8,56,64,... _not in the OEIS_. The start 1,7,9 produces 1,7,9,63,81,... _not in the OEIS_. The start 1,7,10 produces 1,7,10,70,100,700,1000,7000,... _not in the OEIS_. etc. The start 1,8,2 produces https://oeis.org/A164587 [a(n) = 2*a(n - 2) for n > 2; a(1) = 1, a(2) = 8]. The start 1,8,3 produces 1,8,3,24,9,72,27,216,81,648,243,... _not in the OEIS_. The start 1,8,4 produces almost https://oeis.org/A135520 [erase the 1st term, then a(n) = 4*a(n-2)]. The start 1,8,5 produces 1,8,5,40,25,200,125,... _not in the OEIS_. The start 1,8,6 produces 1,8,6,48,36,... _not in the OEIS_. The start 1,8,7 produces 1,8,7,56,49,... _not in the OEIS_. The start 1,8,9 produces 1,8,9,72,81,... _not in the OEIS_. The start 1,8,10 produces 1,8,10,80,100,800,1000,... _not in the OEIS_. etc. The start 1,9,2 produces 1,9,2,18,4,36,8,72,16,144,32,288,... _not in the OEIS_. The start 1,9,3 produces almost https://oeis.org/A162852 [a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1]. But, again, a lot of not-distinct terms. The start 1,9,4 produces 1,9,4,36,16,144,... _not in the OEIS_. The start 1,9,5 produces 1,9,5,45,25,225,... _not in the OEIS_. The start 1,9 6 produces 1,9,6,54,36,... _notin the OEIS_. The start 1,9,7 produces 1,9,7,63,49,... _not in the OEIS_. The start 1,9,8 produces 1,9,8,72,64,... _not in the OEIS_. The start 1,9,10 produces 1,9,10,90,100,900,1000,... _not in the OEIS_. etc. The start 1,10,2 produces 1,10,2,20,4,40,8,80,16,160,32,320,... _not in the OEIS_. The start 1,10,3 produces 1,10,3,30,9,90,27,270,... _not in the OEIS_. etc. The start 2,1,6 produces 2,1,6,3,18,9,54,27,... _not in the OEIS_. As 19 is prime, the start 20,19 produces 20,19,40,38,80,72,... _not in the OEIS_ (and never in the OEIS, I guess). etc. Note that the remark here https://oeis.org/A164073, by a friend of mine ("Absolute second differences are the sequence itself"), is always true for some seqs above that follow the pattern: 1, a, 2, 2a, 4, 4a, 8, 8a, 16, 16a... for a = 0 to k. Oh, and what about picking 5 (instead of 4) successive integers a, b, c, d, e in a seq of distinct terms such that a/b is always equal to d/e? Never mind, this has been already submitted by Paul Curtz: https://oeis.org/A133464 Too good, the OEIS'authors!-)) My two,three, four cents. É. P.-S.#1 Many thanks to Klaus Brockhaus who computed 10 years ago a lot of the above, and forgive the errors I left in this mail, despite my carefull rerereading). P.-S.#2 Let me pick again (in a similar seq T) 5 successive integers a, b, c, d, e -- but such that a/c is now equal to d/e: T = 1, 2, 3, 4, 12, 24, 96, 576,... Ah ah, _T is not in the OEIS_!-)) Best, É.

3 6

[math-fun] Dean Hickerson
by James Propp 21 Oct '19

21 Oct '19

Anyone have an email address for Dean Hickerson? Jim Propp

2 1

[math-fun] Dean Hickerson
by James Propp 21 Oct '19

21 Oct '19

Anyone have a current email address for Dean Hickerson? "dean at math daht ucdavis daht edu" and "dean at ucdmath daht ucdavis daht edu" aren't working anymore. Thanks, Jim Propp

5 5

[math-fun] "LHS Sometimes returns precision errors? What the bleep is it doing numerics for an identity map?"
by Bill Gosper 21 Oct '19

21 Oct '19

Brad, there's a really interesting answer. E.g., In[68]:= Sin[π/34] // FunctionExpand Out[68]= 1/(4 Sqrt[2/( 15 + Sqrt[17] - Sqrt[2 (17 - Sqrt[17])] - Sqrt[ 2 (34 + 6 Sqrt[17] + Sqrt[2 (17 - Sqrt[17])] - Sqrt[ 34 (17 - Sqrt[17])] + 8 Sqrt[2 (17 + Sqrt[17])])])]) (Rather cumbrous, Now undo it.) In[69]:= ArcSin@% // FullSimplify // Timing Out[69]= {0.312125, π/34} How could it take .3 sec to look that up? Alternatively, how could it get π/34 so quickly? Or at all? Here Mathematica recognizes Macsyma's nicer rendition: In[70]:= ArcSin[(-\[Sqrt](\[Sqrt]17 + 3) \[Sqrt](4 \[Sqrt]17 - 2 \[Sqrt](34 - 2 \[Sqrt]17)) + \[Sqrt](34 - 2 \[Sqrt]17) + \[Sqrt]17 - 1)/16] // FullSimplify // Timing Out[70]= {0.948516, π/34} even though it wasn't able to simplify its version to Macsyma's. So where did it get the notion of π/34? It guessed it numerically! It is then fairly routine to show that the two surds are identical, or that Macsyma's satisfies In[72]:= MinimalPolynomial[Sin[π/34], x] // Timing Out[72]= {0.002874, 1 - 8 x - 40 x^2 + 80 x^3 + 240 x^4 - 192 x^5 - 448 x^6 + 128 x^7 + 256 x^8} At least this is how Macsyma did it. Without even requiring a ratsimp or radcan. —rwg

4 4

Re: [math-fun] lituus?
by Bill Gosper 20 Oct '19

20 Oct '19

Thanks Gareth! lituus <http://gosper.org/lituus.gif> Now what's a convincing portrayal of the sector area constancy? Hey Eric, gifs in MathWorld? —rwg On Sun, Oct 20, 2019 at 4:10 PM rwg <rwg(a)ma.sdf.org> wrote: > > > -------- Original Message -------- > Subject: Re: [math-fun] lituus? > Date: 2019-10-17 19:06 > From: Gareth McCaughan <gareth.mccaughan(a)pobox.com> > To: math-fun(a)mailman.xmission.com > Reply-To: math-fun <math-fun(a)mailman.xmission.com> > > On 14/10/2019 12:42, Bill Gosper wrote: > > > MathWorld: > > "The lituus is the locus of the point [image: P] moving such that the > > area > > <http://mathworld.wolfram.com/Area.html> of a circular sector > > <http://mathworld.wolfram.com/CircularSector.html> remains constant." > > What circle? What sector? Pic? —rwg > > It's r^2 theta = const. > > So for a point P at radius r and angle theta, take a circular > sector spanning angles 0..theta and radius r. As the spiral > winds inward around the origin, the radius decreases just enough > to keep that area the same. For "almost all" of the curve we > have theta>2pi and of course we need to count the area with > multiplicity. > > So e.g. if we take the constant to be 1, then at angle 2pi > (i.e., the first time the curve crosses the positive x-axis) > we have r = sqrt(1/2pi) and that circle is covered just once; > at angle 4pi where it crosses again we have r = sqrt(1/4pi) > so the circle is sqrt(2) times smaller in radius but is covered > twice. >

1 0

[math-fun] A Quercus point with Geo-location
by Brad Klee 20 Oct '19

20 Oct '19

According to science news articles, in order to accomplish their yearly dismissals, deciduous trees have specially-evolved abscission cells. The release of used-up leafs is not a passive firing, rather it is an active process involving climate-dependent chemical signals. Naively we could hypothesize that the spatial distribution of active abscission cells is uniform random on the entire canopy volume at every instant of time; however, a walk around the block, and I could easily find many counterexamples--especially oaks, including the Pin Oak (Quercus Palustris). Here is a picture taken today, of tree T1438 (with a day moon in the background, and now with GPS metadata, cool!): https://0x0.st/z3Ma.jpg https://0x0.st/z3Mm.png Most of the tree remains green, but already a few branches have gone entirely bare. Then there are a few working groups of mostly bronze color, which seem likely to fall soon. This data suggests that distribution function for chemical signalling is highly correlated on the small scale of a single leaf, but remains at least partially random on the large scale of the entire tree. It seems impossible to derive chemical equations of motion and the ensuing statistics, but still I am curious, is there a better available empirical explanation as to what the signals are, where they come from, and how they flow through the tree's network of branches? And how much do these effects depend on species? --Brad [1] https://www.npr.org/templates/story/story.php?storyId=114288700 [2] https://www.esf.edu/pubprog/brochure/leaves/leaves.htm

1 0

[math-fun] Fwd: Re: "LHS Sometimes returns precision errors? What the bleep is it doing numerics for an identity map?"
by Bill Gosper 19 Oct '19

19 Oct '19

Date: 2019-10-19 09:23 From: R B <bossipm(a)outlook.com> Some googling gave me https://nanopdf.com/download/sines-and-cosines-for-fractions-of_pdf from which Sin[π/34] can be derived simply. Arie Bos "So, sin π/n and cos π/n can be found for n = 2k ×3p ×5q ×17r ×257s ×65537t where k is a positive or zero integer and the indices p, q, r, s, t are all either zero or 1." RubbEESSHimo! In[90]:= #@Sin[π/7] & /@ {ToRadicals, Developer`TrigToRadicals} During evaluation of In[90]:= Developer`TrigToRadicals::obs: Developer`TrigToRadicals has been superseded by ToRadicals, [I believe this to be completely false.] and is now obsolete. It will not be included in future versions of the Wolfram Language. [And I pray this will not happen.] Out[90]= {-(1/2) (-1)^(5/14) (-1 + (-1)^(2/7)), -(1/2) I (1/3 (1/ 2 (1 - I Sqrt[7]) - ((-1 + I Sqrt[3]) (1/2 (-1 + I Sqrt[7]) + 1/2 (-1 - I Sqrt[7]) (1/2 (-1 + I Sqrt[3]) + 1/4 (-1 + I Sqrt[3])^2)))/( 2 (6 + 3/4 (-1 + I Sqrt[3]) (-1 + I Sqrt[7]) + 1/2 (-1 - I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)) - 1/4 (-1 + I Sqrt[3])^2 (6 + 3/4 (-1 + I Sqrt[3]) (-1 + I Sqrt[7]) + 1/2 (-1 - I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)) + 1/3 (1/2 (-1 - I Sqrt[7]) + ((-1 + I Sqrt[3])^2 (1/2 (-1 - I Sqrt[7]) + 1/2 (-1 + I Sqrt[7]) (1/2 (-1 + I Sqrt[3]) + 1/4 (-1 + I Sqrt[3])^2)))/( 4 (6 + 3/4 (-1 + I Sqrt[3]) (-1 - I Sqrt[7]) + 1/2 (-1 + I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)) + 1/2 (-1 + I Sqrt[3]) (6 + 3/4 (-1 + I Sqrt[3]) (-1 - I Sqrt[7]) + 1/2 (-1 + I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)))} In[325]:= MinimalPolynomial@% Out[325]= {-7 + 56 #1^2 - 112 #1^4 + 64 #1^6 &, -7 + 56 #1^2 - 112 #1^4 + 64 #1^6 &} In[328]:= %%%[[1]]@Sin[π/7] // FullSimplify Out[328]= 0 How can that bullbleep page remain up? —rwg > -----Oorspronkelijk bericht----- > Van: math-fun <math-fun-bounces(a)mailman.xmission.com> Namens Bill > Gosper > Verzonden: zaterdag 19 oktober 2019 07:46 > Aan: math-fun(a)mailman.xmission.com > CC: Wolfram Technical Support <support(a)wolfram.com> > Onderwerp: [math-fun] "LHS Sometimes returns precision errors? What the > bleep is it doing numerics for an identity map?" > Urgentie: Hoog > > Brad, there's a really interesting answer. E.g., In[68]:= Sin[π/34] // > FunctionExpand > > Out[68]= 1/(4 Sqrt[2/( > 15 + Sqrt[17] - Sqrt[2 (17 - Sqrt[17])] - Sqrt[ > 2 (34 + 6 Sqrt[17] + Sqrt[2 (17 - Sqrt[17])] - Sqrt[ > 34 (17 - Sqrt[17])] + 8 Sqrt[2 (17 + Sqrt[17])])])]) > > (Rather cumbrous, Now undo it.) > > In[69]:= ArcSin@% // FullSimplify // Timing > > Out[69]= {0.312125, π/34} > > How could it take .3 sec to look that up? Alternatively, how could it > get π/34 > so quickly? Or at all? > > Here Mathematica recognizes Macsyma's nicer rendition: > In[70]:= ArcSin[(-\[Sqrt](\[Sqrt]17 + 3) \[Sqrt](4 \[Sqrt]17 - > 2 \[Sqrt](34 - 2 \[Sqrt]17)) + \[Sqrt](34 - > 2 \[Sqrt]17) + \[Sqrt]17 - 1)/16] // FullSimplify // Timing > > Out[70]= {0.948516, π/34} > > even though it wasn't able to simplify its version to Macsyma's. So > where did > it get the notion of π/34? It guessed it numerically! It is then > fairly routine to > show that the two surds are identical, or that Macsyma's satisfies > In[72]:= > MinimalPolynomial[Sin[π/34], x] // Timing > > Out[72]= {0.002874, > 1 - 8 x - 40 x^2 + 80 x^3 + 240 x^4 - 192 x^5 - 448 x^6 + 128 x^7 + > 256 x^8} > > At least this is how Macsyma did it. Without even requiring a ratsimp > or > radcan. > —rwg > _______________________________________________ > math-fun mailing list

2 1

[math-fun] Zulu Bungy was tragically put out of business (collapse side panel)
by Bill Gosper 19 Oct '19

19 Oct '19

https://www.google.com/maps/place/37%C2%B056'05.0%22N+22%C2%B059'02.0%22E/@37.9217471,22.9981443,557a,35y,38.17t/data=!3m1!1e3!4m5!3m4!1s0x0:0x0!8m2!3d37.934722!4d22.983889?hl=en when the state-owned cement company substituted spanakopita flour during a shortage. Seriously, can anyone explain this effect? —rwg

3 2

[math-fun] FullSimplify[ArcTan[Cos[x], Sin[x]]] == ArcTan[Cos[x], Sin[x]] ?
by Brad Klee 18 Oct '19

18 Oct '19

Ouch! I think it's just a modulus function, that's what the graph looks like at least: Show[ Plot[ArcTan[Cos[x], Sin[x]], {x, -10 Pi, 10 Pi}, PlotStyle -> Thick], Plot[Mod[x + Pi, 2 Pi] - Pi, {x, -10 Pi, 10 Pi}, PlotStyle -> Directive[Dashing[.05], Thick, Red]], PlotRange -> All, ImageSize -> 500 ] Or is it supposed to be: Show[ Plot[ArcTan[Cos[x], Sin[x]], {x, -10 Pi, 10 Pi}, PlotStyle -> Thick], Plot[Pi - Mod[Pi - x, 2 Pi], {x, -10 Pi, 10 Pi}, PlotStyle -> Directive[Dashing[.05], Thick, Red]], PlotRange -> All, ImageSize -> 500] Gotta watch those boundary values, and don't forget all of the broken evaluation over complexes: ArcTan[Cos[x],Sin[x]]==(Pi-Mod[Pi-Re[x],2*Pi]+I*Im[x]) LHS Sometimes returns precision errors? What the bleep is it doing numerics for an identity map? Try: ArcTan[Cos[I #], Sin[I #]] & /@ Range[100] ArcTan[Sin[I #]/Cos[I #]] & /@ Range[100] Refine[ArcTanh[Tanh[x]], {x \[Element] Reals}] Surprisingly, the second alternative does quite a bit better, managing to get up to n=75 rather than n=18, but still dies at 76 and thereafter. This is okay as long as we only need a thin strip around the real axis, but in general, no, "Indeterminate" is not a good return. --Brad

3 2

[math-fun] lituus?
by Bill Gosper 17 Oct '19

17 Oct '19

MathWorld: "The lituus is the locus of the point [image: P] moving such that the area <http://mathworld.wolfram.com/Area.html> of a circular sector <http://mathworld.wolfram.com/CircularSector.html> remains constant." What circle? What sector? Pic? —rwg

2 1