A365515
Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence starting from 0.
Original entry on oeis.org
0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 7, 4, 0, 1, 5, 13, 12, 5, 0, 1, 6, 21, 32, 20, 6, 0, 1, 7, 31, 55, 71, 30, 7, 0, 1, 8, 43, 108, 153, 124, 44, 8, 0, 1, 9, 57, 154, 366, 368, 218, 65, 9, 0, 1, 10, 73, 256, 668, 926, 856, 375, 80, 10, 0, 1, 11, 91, 333, 1153, 2214, 2286, 1424, 572, 96, 11
Offset: 1
Table begins:
n\k | 1 2 3 4 5 6 7 8 9
----+---------------------------------------------------
1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
2 | 0, 1, 3, 7, 12, 20, 30, 44, 65, ...
3 | 0, 1, 4, 13, 32, 71, 124, 218, 375, ...
4 | 0, 1, 5, 21, 55, 153, 368, 856, 1424, ...
5 | 0, 1, 6, 31, 108, 366, 926, 2286, 5733, ...
6 | 0, 1, 7, 43, 154, 668, 2214, 6876, 16864, ...
7 | 0, 1, 8, 57, 256, 1153, 4181, 14180, 47381, ...
8 | 0, 1, 9, 73, 333, 1822, 8043, 28296, 102042, ...
9 | 0, 1, 10, 91, 500, 3119, 13818, 59174, 211135, ...
Cf.
A001477 (n=1),
A025582 (n=2),
A051912 (n=3),
A365300 (n=4),
A365301 (n=5),
A365302 (n=6),
A365303 (n=7),
A365304 (n=8),
A365305 (n=9),
A002061 (k=4),
A369817 (k=5),
A369818 (k=6),
A369819 (k=7),
A347570.
-
from itertools import count, islice, combinations_with_replacement
def A365515_gen(): # generator of terms
asets, alists, klist = [set()], [[]], [0]
while True:
for i in range(len(klist)-1,-1,-1):
kstart, alist, aset = klist[i], alists[i], asets[i]
for k in count(kstart):
bset = set()
for d in combinations_with_replacement(alist+[k],i):
if (m:=sum(d)+k) in aset:
break
bset.add(m)
else:
yield k
alists[i].append(k)
klist[i] = k+1
asets[i].update(bset)
break
klist.append(0)
asets.append(set())
alists.append([])
A365515_list = list(islice(A365515_gen(),30))
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
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.
-
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
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
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.
-
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
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
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.
-
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
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
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.
-
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
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
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.
-
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
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