A101274 a(1)=1; for n>1, a(n) is the smallest positive integer such that the set of all sums of adjacent elements up to and including a(n) contains no number more than once.

1, 2, 4, 5, 8, 10, 14, 21, 15, 16, 26, 25, 34, 22, 48, 38, 71, 40, 74, 90, 28, 69, 113, 47, 94, 54, 46, 143, 153, 83, 128, 49, 249, 75, 133, 225, 125, 131, 270, 145, 230, 199, 237, 206, 201, 299, 136, 346, 72, 272, 120, 55, 453, 247, 376, 427, 124, 535, 87, 242, 431, 283, 227, 212, 940, 318, 387, 311, 391, 325

Offset

1,2

Comments

Does the sequence together with the sums of adjacent elements include all positive integers? Choosing starting values other than a(1)=1 gives other sequences. We could ask, for a given n, which such sequences have the smallest sum of a(k) from k=1 to n.

The first differences of A005282. [Zak Seidov, Nov 06 2010]

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..483

E. Giaquinta and S. Grabowski, New algorithms for binary jumbled pattern matching, arXiv preprint arXiv:1210.6176 [cs.DS], 2012. - From N. J. A. Sloane, Jan 01 2013

Example

a(8)=21 because the set of sums of adjacent elements to this point, call it s(7) is {1,2,3,4,5,6,7,8,9,10,11,12,13,14,17,18,19,20,23,24,27,29,30,32,37,41,43,44}.
The first number missing from this list is 15, but a(8) cannot equal 15 because 15+14=29 and 29 is already in s(7). Similarly a(8) cannot be 16 because 16+14=30.

Mathematica

t = {1}; sms = {2}; k = 1;
Do[k++; While[Intersection[sms, k + t] != {}, k++]; sms = Join[sms, t + k, {2 k}]; AppendTo[t, k], {41}];
Differences[t] (* Jean-François Alcover, Feb 13 2019, after T. D. Noe in A005282 *)

Prog

(Python)
from itertools import count, islice
def A101274_gen(): # generator of terms
    aset1, aset2, alist, n = {1}, set(), [1], 1
    for k in count(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)
A101274_list = list(islice(A101274_gen(), 30)) # Chai Wah Wu, Sep 05 2023

Crossrefs

Cf. A005282.

Sequence in context: A262937 A249508 A163295 * A363446 A080222 A050539

Adjacent sequences: A101271 A101272 A101273 * A101275 A101276 A101277

Keyword

nonn

Author

David S. Newman, Dec 20 2004

Status

approved