December 2005 - mathfun@mailman.xmission.com

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

05 Dec '05

Kerry, You've said some interesting things about fractal sequences, like how to detect them. But what is their underlying definition? Thanks, Dan

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Re: [math-fun] A few self-descriptive fractal sequences
by Kerry Mitchell 05 Dec '05

05 Dec '05

Here are a few more self-descriptive fractal sequences. "Upper trimming" is another technique for determining if a sequence is a fractal sequence, if the sequence contains all positive integers. The first occurrence of each integer is deleted from the sequence. If the resulting sequence is the same is the original, then it is a fractal sequence. This is an example of a fractal sequence: 1, 1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 10, 11, 12, 4, 13, 14, 2, ... If we denote the first occurrence of each integer by X, we get: X, 1, X, 1, X, X, 2, X, 1, X, X, X, 3, X, X, X, X, 4, X, X, 2, ... and dropping the Xs: 1, 1, 2, 1, 3, 4, 2, ... Which is the beginning of the original sequence. This sequence is also self-descriptive, in that each element gives the number of Xs (first occurrences of integers) that were removed just before it. Here are the first 200 terms: 1, 1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 10, 11, 12, 4, 13, 14, 2, 15, 16, 17, 18, 19, 5, 20, 1, 21, 22, 23, 24, 25, 26, 6, 27, 28, 29, 30, 31, 32, 33, 7, 34, 35, 36, 37, 38, 39, 40, 41, 8, 42, 43, 44, 3, 45, 46, 47, 48, 49, 50, 51, 52, 53, 9, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 10, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 11, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 12, 87, 88, 89, 90, 4, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 13, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 14, 118, 119, 2, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 15, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 16, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 17, 168, 169, 170, 171, 172, 173, 174, 175, 176 Here's another one: 1, 1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 4, 6, 3, 5, 7, 2, 8, 1, 4, 6, 3, 9, 5, 7, 2, 10, ... This sequence gives the number of numbers that are retained between Xs that are dropped. Alternatively, each element is the number of numbers between two first occurrences of integers. For example, the first 3 above describes the three numbers 2, 1, 4 between the first 5 and the first 6. Here are the first 200 terms: 1, 1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 4, 6, 3, 5, 7, 2, 8, 1, 4, 6, 3, 9, 5, 7, 2, 10, 8, 1, 4, 6, 3, 11, 9, 5, 12, 7, 13, 2, 10, 8, 1, 14, 4, 6, 3, 11, 9, 5, 15, 12, 7, 13, 16, 2, 10, 8, 1, 14, 17, 4, 6, 3, 11, 9, 5, 15, 18, 12, 7, 19, 13, 16, 2, 10, 8, 1, 14, 17, 20, 4, 21, 6, 3, 11, 9, 22, 5, 15, 18, 12, 7, 19, 23, 13, 16, 2, 24, 10, 8, 1, 14, 17, 20, 4, 21, 6, 25, 3, 11, 9, 22, 5, 26, 15, 18, 12, 7, 19, 23, 13, 27, 16, 2, 28, 24, 10, 8, 1, 14, 17, 20, 4, 21, 6, 29, 25, 3, 11, 9, 22, 5, 26, 15, 30, 18, 31, 12, 7, 19, 23, 32, 13, 27, 16, 2, 28, 24, 33, 10, 8, 1, 34, 14, 17, 20, 4, 21, 6, 29, 25, 3, 11, 9, 35, 22, 5, 26, 15, 30, 18, 31, 12, 7, 36, 19, 23, 32, 13, 27, 37, 16, 2, 28, 24, 33, 10, 8, 1 This last fractal sequence describes the number of Xs that are dropped and the number of numbers written between dropped Xs: 1, 1, 2, 1, 3, 4, 2, 1, 5, 3, 6, 7, 8, 4, 2, 1, 9, 10, 11, 12, 5, 3, 6, 7, 13, 14, 8, 4, 15, 2, 16, 17, 18, 19, 20, 1, 9, 10, 11, 12, 21, 22, 23, 5, 3, 6, 24, 25, 26, 27, 28, 29, 7, 13, 14, 8, 4, 15, 30, 31, 32, 33, 34, 35, 36, 2, 16, 17, 18, 19, 20, 1, 37, 38, 39, 40, 41, 42, 43, 44, 9, 10, 11, 12, 21, 22, 23, 5, 45, 46, 47, 48, 3, 6, 24, 25, 49, 50, 26, 27, 51, 28, 52, 53, 54, 55, 56, 57, 58, 59, 60, 29, 7, 13, 14, 8, 4, 15, 30, 31, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 32, 33, 34, 35, 36, 2, 16, 17, 18, 19, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 20, 1, 37, 38, 39, 40, 41, 42, 43, 44, 9, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 10, 11, 12, 21, 22, 23, 5, 45, 46, 47, 48, 3, 94, 95, 96, 97, 98, 6, 24, 25, 49, 50, 99, 100, 101, 26 Kerry

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