A287178 Smallest number such that the circular adjacent sum set has no multiple entries.

1, 2, 4, 8, 10, 16, 20, 28, 42, 30, 32, 52, 50, 68, 44, 96, 76, 142, 80, 148, 180, 56, 138, 226, 94, 188, 108, 92, 286, 306, 166, 256, 98, 498, 150, 266, 450, 250, 262, 540, 290, 460, 398, 474, 412, 402, 598, 272, 692, 144, 544, 240, 110, 906, 494, 752, 854, 248

Offset

1,2

Comments

Let a1, a2, ..., ak be the first k terms of a sequence. The term "circular adjacent sum set" is the set of sums of 1 to k adjacent terms, where a1 is taken to be adjacent to ak. For example if the first 3 terms of the sequence are 1, 2, and 4 then the circular adjacent sum set is {1,2,4,3,6,5,7}.

Another example. If the first three terms of the sequence are 1,2,3 then the circular adjacent sum set is {1,2,3,3,5,4,7}. One element is repeated because 3 is the sum of adjacent elements in 2 ways, as 3 and as 1+2.

Links

Table of n, a(n) for n=1..58.

Example

For n=4, and a(1)=1, a(2)=2, and a(3)=4. a(4) cannot be 1, 2, or 3 because then there would be two different sums equal to 3. a(4) cannot be 5 because 5+1=2+4. It cannot be 6 because 6=2+4. It cannot be 7 because 7=4+2+1. For a(4)=8 the circular adjacent sum set is {1,2,4,8,3,6,12,9,7,14,13,11,15}. All 13 of these sums are different, so a(4) is 8.

Mathematica

ok[v_] := Block[{w = v, n = Length@v}, w=Reap[Do[w = RotateLeft[w]; Do[Sow[ Total@ Take[w, j]], {j, n - 1}], {n}]][[2, 1]]; Length[w] == Length@ Union@ w]; a = {1}; While[Length[a] < 30, AppendTo[a, 2]; While[! ok[a], a[[-1]]++]]; a (* Giovanni Resta, May 30 2017 *)

Prog

(Python)
a, isums, lsums, rsums, xsums = [], set(), set([0]), set([0]), set([0])
for i in range(100):
    for new in range(1, sum(a)+2):
        nrsums, nxsums = set([0]), set([0])
        for x in rsums:
            xn = x+new
            if xn in isums or xn in lsums:
                break
            nrsums.add(xn)
        else:
            for x in xsums.union(lsums):
                xn = x+new
                if xn in isums or xn in rsums or xn in lsums:
                    break
                nxsums.add(xn)
            else:
                a.append(new)
                isums.update(rsums)
                xsums, rsums = nxsums, nrsums
                lsums.add(sum(a))
                break
print(a)
# Andrey Zabolotskiy, May 30 2017
(Python)
from itertools import count, islice
def A287178_gen(): # generator of terms
    aset1, aset2, alist, n = set(), set(), [], 0
    for k in count(1, 2):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k-n
                n = k
                alist.append(k)
                aset1.add(k)
                aset2.update(bset2)
A287178_list = list(islice(A287178_gen(), 41)) # Chai Wah Wu, Sep 05 2023

Crossrefs

Sequence in context: A060378 A185064 A036975 * A093547 A068382 A356232

Adjacent sequences: A287175 A287176 A287177 * A287179 A287180 A287181

Keyword

nonn

Author

David S. Newman, May 21 2017

Extensions

More terms from Andrey Zabolotskiy, May 30 2017

Status

approved