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A287178 Smallest number such that the circular adjacent sum set has no multiple entries. 0
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 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Sequence in context: A060378 A185064 A036975 * A093547 A068382 A025612

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

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Last modified September 30 00:37 EDT 2020. Contains 337432 sequences. (Running on oeis4.)