This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Kevin O'Bryant has authored 23 sequences. Here are the ten most recent ones:
For n=3, the a(3) = 3 pairs are (5,6), (6,6), (6,5). For n=4, the a(4) = 10 pairs are (u,10), (u,9), (10,8), (10,7), where 7 <= u <= 10.
For n=3 the three weak orderings are: 11<12=21<13=31=22<23=32<33, 11<12=21<22<13=31<23=32<33, and 11<12=21<13=31<22<23=32<33, which arise from the sets {1,2,3}, {1,2,4}, and {1,3,4}, respectively.
For n=4, there are 39 weak orderings, but only 25 arise from sets of real numbers. For example, the weak ordering 11<12=21<13=31<22<14=41=23=32<24=42=33<34=43<44 does not arise from any set of real numbers.
For n=2 the a(2)=1 ordered partition is {(1,1)}<{(2,1),(1,2)}<{(2,2)}. We can encode this as 11<12<22, writing "ij" for the pair (i,j).
For n=3 one of the a(3)=3 ordered partitions is {(1,1)}<{(1,2),(2,1)}<{(1,3),(3,1),(2,2)}<{(2,3),(3,2)}<{(3,3)}, which is encoded as either 11<12<13=22<23<33 or 11<12<22=13<23<33. The other two ordered partitions can be encoded as 11<12<22<13<23<33 and 11<12<13<22<23<33.
From _Andrew Howroyd_, Sep 15 2024: (Start)
The a(3) = 3 symmetric matrices are:
[1 2 3] [1 2 3] [1 2 4]
[2 3 4] [2 4 5] [2 3 5]
[3 4 5] [3 5 6] [4 5 6]
(End)
\\ See PARI link in A374514 for program code. vector(10, n, A376162(n)) \\ Andrew Howroyd, Sep 16 2024
a(2) = 30, as all 28 nonincreasing sums from {0,1,3,7,12,20,30}, namely 0+0 < 0+1 < 1+1 < ... < 7+20 < 0+30 < 1+30 < 12+20 <3+30 < 7+30 < 20+20 < 12+30 < 20+30 < 30+30, are distinct, and all other 7-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20,30}.
# uses Python code from A369818 from itertools import count, combinations_with_replacement def A369819(n): alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1),A369818(n)] 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(alist[-1]+1): 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
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}.
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(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}.
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
a(4) != 70 because 70+1+1+0+0+0+0+0 = 9+9+9+9+9+9+9+0.
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(3) != 7 because 7+0+0+0+0+0+0 = 1+1+1+1+1+1+1.
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(4) != 72 because 72+1+1+1+1+1+1+1+1+0 = 10+10+10+10+10+10+10+10+0.
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(5) != 50 because 50+1+1+1+1+0 = 43+7+1+1+1+1.
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
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