A365515 -id:A365515 - cp's OEIS frontend

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

Dan Hoey

nonn

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

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

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.

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.

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

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

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

A051912 a(n) is the smallest integer such that the sum of any three ordered terms a(k), k <= n, is unique.

Original entry on oeis.org

0, 1, 4, 13, 32, 71, 124, 218, 375, 572, 744, 1208, 1556, 2441, 3097, 4047, 5297, 6703, 7838, 10986, 12331, 15464, 19143, 24545, 28973, 34405, 37768, 45863, 50876, 61371, 68302, 77917, 88544, 101916, 122031, 131624, 148574, 171236, 197814

Offset: 0

Wouter Meeussen, Dec 17 1999

			Three terms chosen from {0,1,4} can be 0+0+0; 0+0+1; 0+1+1; 1+1+1; 0+0+4; 0+1+4; 1+1+4; 0+4+4; 1+4+4; 4+4+4 are all distinct (3*4*5/6 = 10 terms), so a(2) = 4 is the next integer of the sequence after 0 and 1.

Chai Wah Wu, Table of n, a(n) for n = 0..225 (terms 0..100 from Robert Israel)

Row 3 of A365515.

Cf. A025582, A036241, A062065.

A[0]:= 0: S:= {0}: S2:= {0}: S3:= {0}:
for i from 1 to 40 do
  for x from A[i-1] do
    if (map(t -> t+x,S2) intersect S3 = {}) and (map(t -> t+2*x, S) intersect S3 = {}) then
     A[i]:= x;
     S3:= S3 union map(t -> t+x,S2) union map(t -> t+2*x, S) union {3*x};
     S2:= S2 union map(t -> t+x, S) union {2*x};
     S:= S union {x};
     break
    fi
  od
od:
seq(A[i],i=0..40); # Robert Israel, Jul 01 2019

a[0] = 0; a[1] = 1; a[n_] := a[n] = For[A0 = Array[a, n, 0]; an = a[n-1] + 1, True, an++, A1 = Append[A0, an]; A2 = Flatten[Table[A1[[{i, j, k}]], {i, 1, n+1}, {j, i, n+1}, {k, j, n+1}], 2]; A3 = Sort[Total /@ A2]; If[Length[A3] == Length[Union[A3]], Return[an]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 38}] (* Jean-François Alcover, Nov 24 2016 *)

from itertools import count, islice
def A051912_gen(): # generator of terms
    aset1, aset2, aset3, alist = set(), set(), set(), []
    for k in count(0):
        bset2, bset3 = {k<<1}, {3*k}
        if 3*k not in aset3:
            for d in aset1:
                if (m:=d+(k<<1)) in aset3:
                    break
                bset2.add(d+k)
                bset3.add(m)
            else:
                for d in aset2:
                    if (m:=d+k) in aset3:
                        break
                    bset3.add(m)
                else:
                    yield k
                    alist.append(k)
                    aset1.add(k)
                    aset2 |= bset2
                    aset3 |= bset3
A051912_list = list(islice(A051912_gen(),20)) # Chai Wah Wu, Sep 01 2023

More terms from Naohiro Nomoto, Jul 22 2001

A365300 a(n) is the smallest nonnegative integer such that the sum of any four ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 5, 21, 55, 153, 368, 856, 1424, 2603, 4967, 8194, 13663, 22432, 28169, 47688, 65545, 96615, 146248, 202507, 266267, 364834, 450308, 585328, 773000, 986339, 1162748, 1472659, 1993180, 2275962, 3012656, 3552307, 4590959, 5404183, 6601787, 7893270, 9340877

Offset: 1

Kevin O'Bryant, Aug 31 2023

nonn

This is the greedy B_4 sequence.

			a(4) != 12 because 12+1+1+1 = 5+5+5+0.

Chai Wah Wu, Table of n, a(n) for n = 1..50
J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Row 4 of A365515.

Cf. A025582, A051912, A365301, A365302, A365303, A365304, A365305.

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(4,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365300_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],3):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365300_list = list(islice(A365300_gen(),20)) # Chai Wah Wu, Sep 01 2023

a(27)-a(37) from Chai Wah Wu, Sep 01 2023

A365301 a(n) is the smallest nonnegative integer such that the sum of any five ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 6, 31, 108, 366, 926, 2286, 5733, 12905, 27316, 44676, 94545, 147031, 257637, 435387, 643320, 1107715, 1760092, 2563547, 3744446, 5582657, 8089160, 11373419, 15575157, 21480927, 28569028, 40893371, 53425354, 69774260, 93548428, 119627554

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_5 sequence.

			a(4) != 20 because 20+1+1+1+1 = 6+6+6+6+0.

J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Row 5 of A365515.

Cf. A025582, A051912, A365300, A365302, A365303, A365304, A365305.

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(5,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365301_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],4):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365301_list = list(islice(A365301_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(18)-a(28) from Chai Wah Wu, Sep 01 2023
a(29)-a(30) from Chai Wah Wu, Sep 10 2023
a(31)-a(32) from Chai Wah Wu, Mar 02 2024

A365302 a(n) is the smallest nonnegative integer such that the sum of any six ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 7, 43, 154, 668, 2214, 6876, 16864, 41970, 94710, 202027, 429733, 889207, 1549511, 3238700, 5053317, 8502061, 15583775, 25070899, 40588284, 63604514

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_6 sequence.

			a(5) != 50 because 50+1+1+1+1+0 = 43+7+1+1+1+1.

J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Row 6 of A365515.

Cf. A025582, A051912, A365300, A365301, A365303, A365304, A365305.

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(6,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365302_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],5):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365302_list = list(islice(A365302_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(15)-a(19) from Chai Wah Wu, Sep 01 2023

a(20)-a(22) from Chai Wah Wu, Sep 09 2023

A365303 a(n) is the smallest nonnegative integer such that the sum of any seven ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 8, 57, 256, 1153, 4181, 14180, 47381, 115267, 307214, 737909, 1682367, 3850940, 8557010, 18311575, 37925058, 61662056

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_7 sequence.

			a(3) != 7 because 7+0+0+0+0+0+0 = 1+1+1+1+1+1+1.

J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Row 7 of A365515.

Cf. A025582, A051912, A365300, A365301, A365302, A365304, A365305.

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(7,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365303_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],6):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365303_list = list(islice(A365303_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(13)-a(18) from Chai Wah Wu, Sep 13 2023

A365304 a(n) is the smallest nonnegative integer such that the sum of any eight ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 9, 73, 333, 1822, 8043, 28296, 102042, 338447, 1054824, 2569353, 6237718, 15947108, 36179796

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_8 sequence.

			a(4) != 70 because 70+1+1+0+0+0+0+0 = 9+9+9+9+9+9+9+0.

J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Row 8 of A365515.

Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365305.

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(8,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365304_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],7):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365304_list = list(islice(A365304_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(11)-a(15) from Chai Wah Wu, Sep 13 2023

A365305 a(n) is the smallest nonnegative integer such that the sum of any nine ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 10, 91, 500, 3119, 13818, 59174, 211135, 742330, 2464208, 7616100, 19241477, 56562573

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_9 sequence.

			a(4) != 72 because 72+1+1+1+1+1+1+1+1+0 = 10+10+10+10+10+10+10+10+0.

J. Cilleruelo and J. Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Row 9 of A365515.

Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365304.

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(9,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365305_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],8):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365305_list = list(islice(A365305_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(11)-a(14) from Chai Wah Wu, Sep 13 2023

A369817 The fifth term of the greedy B_n set of natural numbers.

Original entry on oeis.org

4, 12, 32, 55, 108, 154, 256, 333, 500, 616, 864, 1027, 1372, 1590, 2048, 2329, 2916, 3268, 4000, 4431, 5324, 5842, 6912, 7525, 8788, 9504, 10976, 11803, 13500, 14446, 16384, 17457, 19652, 20860, 23328, 24679, 27436, 28938, 32000, 33661, 37044, 38872, 42592, 44595, 48668, 50854, 55296, 57673, 62500, 65076

Offset: 1

Kevin O'Bryant, Feb 02 2024

{0, 1, n+1, n^2+n+1, a(n)} is the lexicographically first set of 5 nonnegative integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.

			a(2) = 12, as all 15 nonincreasing sums from {0,1,3,7,12}, namely 0+0 < 0+1 < 1+1 < 0+3 < 1+3 < 3+3 < 0+7 < 1+7 < 3+7 < 0+12 < 1+12 < 7+7 < 3+12 < 7+12 < 12+12, are distinct, and all other 5-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12}.

M. B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
M. B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, B_h-sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Column 5 of A365515.

a[n_] := Floor[(n + 3)/2] n^2 + Floor[(3 n + 2)/2]

def A369817(n): return (n+3>>1)*n**2+(3*n+2>>1) # Chai Wah Wu, Feb 28 2024

a(n) = floor((n + 3)/2) * n^2 + floor((3*n + 2)/2), proved in arXiv:2311.14021.
G.f.: x*(-x^6 + x^5 + 5*x^4 - x^3 + 8*x^2 + 8*x + 4)/((x - 1)*(x^2 - 1)^3). - Chai Wah Wu, Feb 28 2024
E.g.f.: ((2 + 7*x + 5*x^2 + x^3)*cosh(x) + (1 + 6*x + 6*x^2 + x^3)*sinh(x) - 2)/2. - Stefano Spezia, Mar 09 2024

A369818 The sixth term of the greedy B_n set of natural numbers.

Original entry on oeis.org

5, 20, 71, 153, 366, 668, 1153, 1822, 3119, 4448, 6348, 8559, 11565, 14976, 21023, 26220, 33066, 40306, 49601, 59354, 76031, 89248, 106008, 122909, 143989, 165196, 200759, 227660, 261030, 293736, 333825, 373110, 438191, 485952, 544356, 600523, 668573, 734072, 841679, 918988, 1012578, 1101374, 1208065, 1309426, 1474943, 1592000, 1732656

Offset: 1

Kevin O'Bryant, Feb 03 2024

nonn

{0, 1, n+1, n^2+n+1, A369817(n), a(n)} is the lexicographically first set of 6 nonnegative integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.

			a(2) = 20, as all 21 nonincreasing sums from {0,1,3,7,12,20}, namely 0+0 < 0+1 < 1+1 < 0+3 < 1+3 < 3+3 < 0+7 < 1+7 < 3+7 < 0+12 < 1+12 < 7+7 < 3+12 < 7+12 < 0+20 < 1+20 < 3+20 < 12+12 < 7+20 < 12+20 < 20+20, are distinct, and all other 6-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20}.

M. B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
M. B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, B_h-sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.

Column 6 of A365515.

Cf. A369817.

from itertools import count, combinations_with_replacement
def A369818(n):
    alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1)]
    aset = set(sum(d) for d in combinations_with_replacement(alist,n))
    blist = []
    for i in range(n):
        blist.append(set(sum(d) for d in combinations_with_replacement(alist,i)))
    for k in count(max(alist[-1]+1,(n**3>>1)*(1+(n>>2)))):
        for i in range(n):
            if any((n-i)*k+d in aset for d in blist[i]):
                break
        else:
            return k # Chai Wah Wu, Feb 28 2024

Conjectured that a(6n+i) is a quartic polynomial sequence with lead term (1/3)n^4 for each i in {1,2,3,5,6,10} in arxiv:2312.10910.

Proved that (1/8)*n^4 + (1/2)*n^3 <= a(n) <= 0.406671*n^4 + O(n^3) in arxiv:2312.10910.

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