A025582 A B_2 sequence: a(n) is the least value such that sequence increases and pairwise sums of elements are all distinct.

0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, 122, 147, 181, 203, 251, 289, 360, 400, 474, 564, 592, 661, 774, 821, 915, 969, 1015, 1158, 1311, 1394, 1522, 1571, 1820, 1895, 2028, 2253, 2378, 2509, 2779, 2924, 3154, 3353, 3590, 3796, 3997, 4296, 4432, 4778, 4850

Offset: 1

Author

Dan Hoey

Keywords

• nonn

Comments

a(n) is also the least value such that sequence increases and pairwise differences of distinct elements are all distinct.

Examples

			After 0, 1, a(3) cannot be 2 because 2+0 = 1+1, so a(3) = 3.
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..1000
• Alon Amit, What are some interesting proofs using transfinite induction?, Quora, Nov 15 2014.
• LiJun Zhang, Bing Li, and LeeTang Cheng, Constructions of QC LDPC codes based on integer sequences, Science China Information Sciences, June 2014, Volume 57, Issue 6, pp 1-14.
• Eric Weisstein's World of Mathematics, B2 Sequence.
• Index entries for B_2 sequences.

Crossrefs

Row 2 of A365515.

See A011185 for more information.

A010672 is a similar sequence, but there the pairwise sums of distinct elements are all distinct.

Programs

Python

from itertools import count, islice
def A025582_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in count(0):
        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
                alist.append(k)
                aset1.add(k)
                aset2 |= bset2
A025582_list = list(islice(A025582_gen(),20)) # Chai Wah Wu, Sep 01 2023

Sage

def A025582_list(n):
    a = [0]
    psums = set([0])
    while len(a) < n:
        a += [next(k for k in IntegerRange(a[-1]+1, infinity) if not any(i+k in psums for i in a+[k]))]
        psums.update(set(i+a[-1] for i in a))
    return a[:n]
print(A025582_list(20))
# D. S. McNeil, Feb 20 2011

Formula

a(n) = A005282(n) - 1. - Tyler Busby, Mar 16 2024