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Re: [math-fun] A few self-descriptive fractal sequences
by dasimov＠earthlink.net 07 Dec '05

07 Dec '05

Kerry writes: << . . . [several interesting examples of "fractal" sequences, and] . . . Since the exact manner in which a sequence can contain copies of itself varies, I don't know that there is an exact definition of a fractal sequence. >> Hmm. Suppose we take a random sequence of 0's and 1's. With probability 1, every initial string of this sequence is later repeated, nay, repeated infinitely often. Does that mean that except for a set of measure zero (in the product space {0,1}^N endowed with the product measure {1/2,1/2}^N), every sequence is fractal? (N is your favorite definition of the natural numbers.) --Dan

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[math-fun] Re: ellipse packing
by Dean Hickerson 07 Dec '05

07 Dec '05

Richard Schroeppel asked: > Puzzle: Choose some specific ellipse, say a=2 b=1. > Find any packing that has a higher packing density > than circles in the plane (the simple hex packing). And I generalized the question to: > Suppose we're allowed to use more than one shape of ellipse, as long as > all of them have the same area. Can we get a higher density than the > hex packing of circles? I asked Sherman Stein, who asked Don Chakerian, who found a reference that gives negative answers to both questions: > I found in Fejes-Toth's Lagerungen in der Ebene, auf der Kugel, und im > Raum, a proof that the density of the closest packing of plane convex > sets all congruent to a given set K is at most area(K)/area(H), where H > is a hexagon of least area circumscribed about K. Then on p.88 he > remarks that his proof extends to equiareal affine images of a given > convex set, from which one deduces that ellipses of equal area cannot be > packed denser than congruent circular disks. I don't have access to a good math library, so I won't be able to look for the book. Dean Hickerson dean(a)math.ucdavis.edu

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RE: [math-fun] Sum of last ten digits
by Eric Angelini 07 Dec '05

07 Dec '05

Beautiful, Michael -- I had a conversation this morning about this generalization with my friend Alexandre -- you did the job so quickly ! I'm more than delighted ! 1000 thanks ! Best, É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be(a)mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Michael D Beeler Envoyé : mercredi 7 décembre 2005 17:45 À : math-fun(a)mailman.xmission.com Objet : Re: [math-fun] Sum of last ten digits We can generalize the question by using the last s "digits" in radix b, starting witn s-1 zero terms followed by a 1. Of course this still loops because there are b^s possible states of the last s digits. In fact, the loop is usually much smaller: s=2 s=3 s=4 s=5 s=6 s=7 s=8 s=9 s=10 b=2 3 3 1 3 2 1 1 1 5 b=3 4 4 3 7 5 13 14 ? ? b=4 3 3 4 20 13 ? ? ? ? b=5 5 5 1 43 ? ? ? ? ? b=6 3 12 15 110 31 ? ? ? ? b=7 4 9 6 34 13 33 ? ? ? b=8 6 6 45 84 41 249 171 ? ? b=9 9 30 14 52 16 74 20 14654 ? b=10 8 4 50 171 14 461 78 12203 312 When s > b, terms can be more than 2 digits. My program doesn't work for such cases, hence the several "?" for loop length, and some in the upper right half that I filled in by hand. The length of the tail from 0 ... 0 1 to the loop can be definied different ways, depending on where you start and whether you count terms or the s-digit window state. But the longest tail among the cases above is s=9, b=10; its tail is 14244 window-states with 0...01 as state 1. The largest tail-to-loop ratio is s=4, b=5; tail is 20, for a ratio of 20:1. The runner-up is s=7, b=7, ratio 602/33 = ~18.2. The shortest tail is 1, for s=2 and b = 2, 3, 5, 8 and 9. Eric's original problem, s=10, b=10, has a tail of only 13, for the smallest tail-to-loop ratio (1:24) of the cases above. -- Mike _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Sum of last ten digits
by N. J. A. Sloane 07 Dec '05

07 Dec '05

Eric, I'm adding this: %I A112402 %S A112402 0,0,0,0,0,0,0,0,0,1,1,2,4 %N A112402 Next term is the sum of the last 10 digits in the sequence. %C A112402 Digits, not terms! %C A112402 There are only 10^10 possibilities for the last 10 digits, so the sequence must eventually cycle. %O A112402 0,12 %K A112402 nonn,base,more %A A112402 Eric Angelini, Dec 06 2005 Neil

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RE: [math-fun] Sum of last ten digits
by Eric Angelini 07 Dec '05

07 Dec '05

Thanks Neil -- but the seq. was accepted by your database robot a couple of hours ago ! See : A112395 Best, É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be(a)mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de N. J. A. Sloane Envoyé : mercredi 7 décembre 2005 6:01 À : seqfan(a)ext.jussieu.fr; math-fun; math-fun Objet : Re: [math-fun] Sum of last ten digits Eric, I'm adding this: %I A112402 %S A112402 0,0,0,0,0,0,0,0,0,1,1,2,4 %N A112402 Next term is the sum of the last 10 digits in the sequence. %C A112402 Digits, not terms! %C A112402 There are only 10^10 possibilities for the last 10 digits, so the sequence must eventually cycle. %O A112402 0,12 %K A112402 nonn,base,more %A A112402 Eric Angelini, Dec 06 2005 Neil _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] 2-digits sum loop
by Eric Angelini 06 Dec '05

06 Dec '05

One word, Mike : perfect ! Best, É. (my longest ending tail -- starting the loop with 5 : 5 3 2 11 6 5 14 9 5 4 13 8 5 3 2 11 6 5 14 9 5 4 13 7 6 15 8 7 16 9 7 25 END ) ^^^^^^ ^^^^^^ -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be(a)mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Mike Stay Envoyé : mardi 6 décembre 2005 18:01 À : math-fun Objet : Re: [math-fun] 2-digits sum loop > Hello, > consider this « sequence » : > > 17, 9, 8, 17, 9, 8, 17, 9, 8, 17, 9, 8, ... > > ... it's property is that any n is the sum of the two > digits immediately following n. > > ... the seq. is built on the repeated pattern : [17, 9, 8] > > Find another such seq. built on a different pattern. There's the nice sequence 0,0,0,... ;) Sequences should really be built in reverse, since then the next term is forced. It's convenient to start with the cycle forced by 11: 11,6,5,14,9,5,4,13,8,5,3,2,11 This cycle is also the one forced by 13 and 14. 12 falls into the same cycle immediately: ...,5,3,12 15 and 16 take a little longer: ...,13,7,6,15,8,7,16 17 is the cycle you noted: 17,9,8,17 18 falls into the 11-cycle ...2,1,10,9,18 Whatever number you start with, the sum of the first two digits will be a number less than 19. I think the longest finite tail before reaching a two digit number is starting with 1,0, which gives the 11-cycle. > Best, > É. -- Mike Stay metaweta(a)gmail.com http://math.ucr.edu/~mike _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] 2-digits sum loop
by Eric Angelini 06 Dec '05

06 Dec '05

Hello, consider this « sequence » : 17, 9, 8, 17, 9, 8, 17, 9, 8, 17, 9, 8, ... ... it's property is that any n is the sum of the two digits immediately following n. ... the seq. is built on the repeated pattern : [17, 9, 8] Find another such seq. built on a different pattern. Best, É.

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[math-fun] Lego bricks
by N. J. A. Sloane 06 Dec '05

06 Dec '05

The New Yorker for Apr 27/May 4 1998 has an article by Anthony Lane, The Joy of Bricks, where he says that there are 102981500 ways to combine 6 Lego pieces. This is a sequence that appears to be missing from the OEIS. Can anyone supply the earlier terms? Neil Sloane

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RE: [math-fun] Sum of last ten digits
by Eric Angelini 05 Dec '05

05 Dec '05

> so everything repeats with period at most 90^10. > Am I misunderstanding the question? ... Not at all -- this was my conclusion too -- I just wanted to be sure :-) Best, É.

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[math-fun] Sum of last ten digits
by Eric Angelini 05 Dec '05

05 Dec '05

Start with 0 0 0 0 0 0 0 0 0 1 Next term is the sum of the last 10 digits: 0 0 0 0 0 0 0 0 0 1 1 Next term is the sum of the last 10 digits: 0 0 0 0 0 0 0 0 0 1 1 2 Next term is the sum of the last 10 digits: 0 0 0 0 0 0 0 0 0 1 1 2 4 etc. Does this sequence enter in a loop ? Best,

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