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.
Caleb Stanford has authored 10 sequences.
16 is present because phi(16) = 8 and phi(17) = 16, both powers of two. 17 is not present because phi(17) = 16 but phi(18) = 6, not a power of two.
5 is present because phi(5) = 4 and phi(6) = 2, both powers of two. 15 is present because phi(15) = 8 and phi(16) = 8, both powers of two. 17 is not present because phi(17) = 16 but phi(18) = 6, not a power of two.
For example, here is one such 4 X 4 array:
0001
1111
1010
1100
The following 4 X 5 array is a non-example, as there is no path using only left, right, and downward steps:
10000
10111
11101
00001
For the 37 2 X 3 grids, see A359576.
The following 4 X 5 grid is a counterexample that is counted by A359576 but not by the present sequence:
10000
10111
11101
00001
Notice that there is a path of 1s from the top to the bottom, but only via the upward step detour in the third column. There are 8 such 4 X 5 grids, formed from the above by reflection and by toggling the first row, second column and last row, second to last column.
Table starts:
1 3 7 15 31 63 127 ...
1 7 37 175 781 3367 14197 ...
1 17 197 1985 18621 167337 1461797 ...
1 41 1041 22193 433801 8057625 144762849 ...
1 99 5503 247759 10056087 384409519 ...
1 239 29089 2764991 232777209 ...
1 577 153769 30856705 ...
1 1393 812849 ...
1 3363 ...
1 ...
...
a(2) = 3 because there are 2 nonisomorphic posets on two elements, and both embed into the poset of three elements {a, b, c} with ordering a < b (and other pairs are incomparable).
a(3) = 5 because all posets on three elements can be embedded into {a, b, c, d, e} with ordering a < d, b < c < d, and b < e.
# Find an u-poset that contains all n-posets as induced posets.
def find_universal_poset(n,u):
PP = list(Posets(n))
for U in Posets(u):
ok = True
for P in PP:
if not U.has_isomorphic_subposet(P):
ok = False
break
if ok:
return U
return None
a(3) = 0 because there are no prime graphs on 3 vertices. a(4) = 12 because the only prime graph on 4 vertices is a line (path graph P_4), and there are 12 possible labelings of the path graph.
a(2) = 23 because there are 23 graphs on {x, y, 1, 2} that admit a vertex partition separating x and y, such that each vertex in one half of the partition is adjacent to an even number of vertices in the other half. For instance, the graph with four edges (x,y), (x,1), (y,2), (1,2) admits the partition {{x,2},{y,1}}.
a(n) = sum(k=0, n, prod(i=1, k, 2^(i+1))*prod(i=k+1, n, 1 - 2^i)); \\ Michel Marcus, Sep 11 2015
# a_1(n) and a_2(n) both count the same sequence, in two different ways.
def a_1(n) :
# Output the number of 2-rooted graphs in (a) with n+2 vertices
if n == 0 :
return 1
else :
return 2**((n*n + 3*n) // 2) - (2**n - 1) * a_1(n-1)
def a_2(n) :
# Output the number of 2-rooted graphs in (a) with n+2 vertices
# Formula: \sum_{k=0}^n (\prod_{i=1}^k 2^{i+1}) (\prod_{i=k+1}^n (1 - 2^i))
curr_sum = 0
for k in range(0,n+1) :
curr_prod = 1
for i in range(1,k+1) :
curr_prod *= (2**(i+1))
for i in range(k+1,n+1) :
curr_prod *= (1 - (2**i))
curr_sum += curr_prod
return curr_sum
a(2) = 3 because [1,2], [-1,2], and [1,-2] are sortable to the identity [1,2] using only context-directed reversal moves.
a(2) = 3 because out of the 8 signed permutations of size 2, only [1,2], [1,-2], and [-1,2] are sortable. They each take one cdr move. Three other permutations, [-2,-1], [2,-1], and [-2,1] are not sortable to the identity [1,2] but instead sort to [-2,-1]. Finally, [2,1] and [-1,-2] sort to neither [1,2] nor [-2,-1].
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