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

Author

David S. Newman, Dec 20 2004

Keywords

• nonn

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]

Examples

			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.
		

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

Crossrefs

Cf. A005282.

Programs

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 *)

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