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