David Wilson wrote: << I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858.
Actually A002858 is described in EIS as "Ulam numbers: next is uniquely the sum of 2 earlier terms" It's listed in EIS as beginning 1,2,3,4,6,8,11,13,..., (least such) but one could get a whole nother sequence via 1,2,3,5,8,13,21,34,..., (largest such) or 1,2,3,5,4,9,8,17,..., (largest alternating with least) for instance. ------------------------------------------------------------------------------ --------------- To well-define A002858 perhaps one should define it as "[Sequence beginning with 1,2 whose] next term is *the least* unique sum of 2 previous terms" QUESTION: If "2 previous terms" were replaced with "any number of previous terms" in this definition, would it be a different sequence? --Dan A.
On Tue, 11 Feb 2003 asimovd@aol.com wrote:
David Wilson wrote:
<< I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858.
Actually A002858 is described in EIS as "Ulam numbers: next is uniquely the sum of 2 earlier terms"
It's listed in EIS as beginning 1,2,3,4,6,8,11,13,..., (least such) but one could get a whole nother sequence via 1,2,3,5,8,13,21,34,..., (largest such) or 1,2,3,5,4,9,8,17,..., (largest alternating with least) for instance. ------------------------------------------------------------------------------
--------------- To well-define A002858 perhaps one should define it as
"[Sequence beginning with 1,2 whose] next term is *the least* unique sum of 2 previous terms"
QUESTION: If "2 previous terms" were replaced with "any number of previous terms" in this definition, would it be a different sequence?
--Dan A.
Dan A. makes some interesting observations about these sequences. I have also taken a look at Ulam(1,2). I was interested in how long one has to search for the next term. That is, what is the difference between u(n) and u(n-1). [The sequence of first differences.] This is also in Sloane's OEIS and he says no pattern has been found. However based on looking at the first 2000 terms I boldly conjecture that there are only 4 instances where u(n)-u(n-1) = 1. Also it is striking how many times u(n)-u(n-1) = 2 or 3. Just to emphasize this here's the sequence of first differences with all terms > 3 replaced by `*`: 1, 1, 1, 2, 2, 3, 2, 3, 2, `*`, 2, `*`, 2, `*`, 1, `*`, `*`, `*`, `*`, 3, `*`, `*`, `*`, `*`, 2, 3, `*`, `*`, `*`, `*`, `*`, `*`, 3, `*`, `*`, 2, 3, 2, `*`, `*`, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, `*`, `*`, `*`, 3, `*`, `*`, 2, `*`, 2, `*`, `*`, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, 2, 3, `*`, `*`, `*`, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, 3, `*`, `*`, `*`, `*`, 3, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, `*`, 3, `*`, `*`, 2, 3, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, `*`, `*`, `*`, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, 3, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, `*`, `*`, 2, `*`, `*`, 2, 3, 2, `*`, `*`, `*`, 2, `*`, 3, `*`, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, 3, 2, `*`, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, `*`, `*`, 3, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, 2, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, `*`, 2, `*`, `*`, 3, `*`, `*`, 2, `*`, `*`, `*`, 3, `*`, 2, 3, 2, `*`, 2, `*`, `*`, `*`, `*`, 2, `*`, 3, `*`, 2, `*`, 2, `*`, 2, 3, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, 2, `*`, `*`, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 3, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 3, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 3, `*`, 2, `*`, 3, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 3, 2, `*`, `*`, 2, `*`, `*`, 3, `*`, 2, `*`, `*`, 2, `*`, 3, `*`, 2, 3, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, `*`, `*`, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 3, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, 3, 2, `*`, 2, `*`, 3, 2, `*`, `*`, `*`, 3, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 3, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 3, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 2, 3, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, 3, 2, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, `*`, `*`, `*`, 3, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, 2, 3, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 3, `*`, `*`, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 3, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, `*`, 2, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, 3, `*`, 2, 3, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 3, `*`, `*`, 2, `*`, 2, `*`, 2, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 3, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, `*`, 3, `*`, `*`, 2, 3, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, `*`, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, 3, 2, `*`, 2, 3, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, 3, `*`, 2, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, `*`, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, `*`, 2, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, `*`, `*`, 3, `*`, 2, `*`, 3, 2, `*`, `*`, `*`, `*`, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, `*`, 2, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, `*`, 2, 3, 2, `*`, 2, `*`, `*`, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, `*`, `*`, 2, `*`, 3, `*`, 2, 3, 2, `*`, 2, 3, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, `*`, 3, `*`, 2, `*`, `*`, `*`, 2, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, `*`, `*`, 3, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, 2, 3, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, 2, 3, 2, `*`, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 3, 2, `*`, 2, `*`, `*`, `*`, 2, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, `*`, 3, `*`, 2, `*`, 2, 3, 2, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 3, 2, `*`, 2, `*`, `*`, 2, `*`, `*`, 2, `*`, 2, `*`, 3, 2, `*`, 2, `*`, 2, It is also curious that after 2 and 3 the next most frequent difference is 17. Here's a list of pairs [x,y] where y is the number of times the difference x occurs in the first difference sequence of the first 2000 Ulam(1,2) numbers--sorted by decreasing frequency. [[2, 694], [3, 281], [17, 162], [20, 129], [25, 109], [12, 103], [22, 88], [15, 76], [5, 59], [42, 46], [39, 32], [8, 19], [7, 19], [44, 18], [27, 16], [10, 16], [34, 15], [30, 14], [19, 11], [47, 10], [37, 10], [56, 9], [69, 6], [64, 6], [24, 6], [100, 4], [78, 4], [61, 4], [52, 4], [41, 4], [29, 4], [1, 4], [86, 3], [9, 3], [83, 2], [4, 2], Edwin
----- Original Message ----- From: <asimovd@aol.com> To: <math-fun@mailman.xmission.com> Sent: Tuesday, February 11, 2003 6:58 AM Subject: Re: [math-fun] Ulam(1,2)
David Wilson wrote:
<< I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858.
Actually A002858 is described in EIS as "Ulam numbers: next is uniquely the sum of 2 earlier terms"
It's listed in EIS as beginning 1,2,3,4,6,8,11,13,..., (least such)
This is obviously A002858.
but one could get a whole nother sequence via 1,2,3,5,8,13,21,34,..., (largest such)
Where have I seen these numbers before?
or 1,2,3,5,4,9,8,17,..., (largest alternating with least) for instance.
This is interesting. The previous two definitions force increasing sequences, whereas the alternating definition does not. This leads to the following variants: (1) use numbers not already in the sequence, giving your sequence. 1 2 3 5 4 9 8 17 14 31 15 48 21 79 30 127 40 206 41 333 47 539 53 872 63 1411 66 2283 75 3694 76 5977 82 9671 86 15648 92 25319 99 40967 102 66286 108 107253 115 173539 118 280792 121 454331 131 735123 143 1189454 150 (2) use numbers larger than the previous number, giving 1 2 3 5 6 11 12 23 24 47 48 95 96 191 192 383 384 767 768 1535 1536 3071 3072 6143 6144 12287 12288 24575 24576 49151 49152 98303 98304 196607 196608 393215 393216 786431 786432 1572863 1572864 3145727 3145728 6291455
------------------------------------------------------------------------------ To well-define A002858 perhaps one should define it as
"[Sequence beginning with 1,2 whose] next term is *the least* unique sum of 2 previous terms"
There are many sequences in the OEIS with descriptions like "n has such and such relationship to previous elements" which is understood to mean "n is the smallest element greater than the previous element that has such and such relationship to the previous elements" For instance, you might see "Sums of squares of two earlier elements" which means a(1) = 1, a(n+1) = least k >= a(n) which is a sum of squares of two previous elements. The first description is simpler, and pretty much conveys the meaning under normal assumptions.
I prefer the expanded versions of the definitions. The condensed versions were needed in the printed version, but that was 8 years ago. In the database I think it is better to add the extra words to make the definitions clearer. Feel free to send in clearer defintions where there is ambiguity! Neil Sloane
participants (4)
-
asimovd＠aol.com -
David Wilson -
Edwin Clark -
N. J. A. Sloane