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.

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)

CROSSREFS

KEYWORD

nonn

AUTHOR

David S. Newman, May 21 2017

EXTENSIONS

More terms from Andrey Zabolotskiy, May 30 2017

STATUS

approved