cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A339760 Number of (undirected) Hamiltonian paths in the 2 X n king graph.

Original entry on oeis.org

1, 12, 48, 208, 768, 2752, 9472, 32000, 106496, 351232, 1150976, 3756032, 12222464, 39698432, 128778240, 417398784, 1352138752, 4378591232, 14175698944, 45886734336, 148520304640, 480679821312, 1555633799168, 5034389536768, 16292153131008, 52723609239552, 170619454881792, 552140862914560
Offset: 1

Views

Author

Seiichi Manyama, Dec 16 2020

Keywords

Links

Crossrefs

Row 2 of A350729.
Cf. A308129, A339750, A339761, A339762, A339763.

Programs

  • PARI
    Vec((1 + 6*x - 16*x^2 + 24*x^3 - 16*x^4) / ((1 - 2*x)^2 * (1 - 2*x - 4*x^2)) + O(x^20)) \\ Andrew Howroyd, Jan 17 2022
  • Python
    # Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A(start, goal, n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def B(n, k):
    m = k * n
    s = 0
    for i in range(1, m):
        for j in range(i + 1, m + 1):
            s += A(i, j, n, k)
    return s
def A339760(n):
    return B(n, 2)
print([A339760(n) for n in range(1, 21)])

Formula

Empirical g.f.: x*(1 + 6*x - 16*x^2 + 24*x^3 - 16*x^4) / ((1 - 2*x)^2 * (1 - 2*x - 4*x^2)). - Vaclav Kotesovec, Dec 16 2020
The above formula is correct. - Andrew Howroyd, Jan 17 2022

A339762 Number of (undirected) Hamiltonian paths in the 4 X n king graph.

Original entry on oeis.org

1, 208, 4678, 171592, 4743130, 132202038, 3461461060, 88405359072, 2197293738684, 53565801482634, 1284136961473864, 30365618160010650, 709700882866473654, 16422374051280905778, 376744989106882359402, 8578133199326578887346, 194030408441913214687458
Offset: 1

Views

Author

Seiichi Manyama, Dec 16 2020

Keywords

Links

Crossrefs

Row 4 of A350729.
Cf. A003695, A308129, A339760, A339761, A339763.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A(start, goal, n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def B(n, k):
    m = k * n
    s = 0
    for i in range(1, m):
        for j in range(i + 1, m + 1):
            s += A(i, j, n, k)
    return s
def A339762(n):
    return B(n, 4)
print([A339762(n) for n in range(1, 11)])

A339763 Number of (undirected) Hamiltonian paths in the 5 X n king graph.

Original entry on oeis.org

1, 768, 43676, 4743130, 364618672, 28808442502, 2125185542510, 153198148096800, 10739936528121270, 738599412949227054, 49945111084852186032, 3331294312194018084810, 219599512046978073473186, 14331641424452867055092544, 927231520831830806024847178
Offset: 1

Views

Author

Seiichi Manyama, Dec 16 2020

Keywords

Links

Crossrefs

Row 5 of A350729.
Cf. A003778, A308129, A339760, A339761, A339762.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A(start, goal, n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def B(n, k):
    m = k * n
    s = 0
    for i in range(1, m):
        for j in range(i + 1, m + 1):
            s += A(i, j, n, k)
    return s
def A339763(n):
    return B(n, 5)
print([A339763(n) for n in range(1, 11)])

A350729 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n king graph.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 48, 48, 1, 1, 208, 392, 208, 1, 1, 768, 4678, 4678, 768, 1, 1, 2752, 43676, 171592, 43676, 2752, 1, 1, 9472, 406396, 4743130, 4743130, 406396, 9472, 1, 1, 32000, 3568906, 132202038, 364618672, 132202038, 3568906, 32000, 1
Offset: 1

Views

Author

Andrew Howroyd, Jan 16 2022

Keywords

Examples

			Array begins:
===========================================================
m\n | 1    2      3         4           5             6 ...
----+------------------------------------------------------
  1 | 1    1      1         1           1             1 ...
  2 | 1   12     48       208         768          2752 ...
  3 | 1   48    392      4678       43676        406396 ...
  4 | 1  208   4678    171592     4743130     132202038 ...
  5 | 1  768  43676   4743130   364618672   28808442502 ...
  6 | 1 2752 406396 132202038 28808442502 6544911081900 ...
     ...

Links

Crossrefs

Rows 1..5 are A000012, A339760, A339761, A339762, A339763.
Main diagonal is A308129.
Cf. A272445, A307026, A329118, A339098, A339190.

Formula

T(m,n) = T(n,m).

A339751 Number of (undirected) paths in the 3 X n king graph.

Original entry on oeis.org

3, 235, 5148, 96956, 1622015, 25281625, 375341540, 5384233910, 75321922433, 1034169469257, 13999362291892, 187462552894846, 2489361245031701, 32843155609675341, 431132757745615932, 5637280548371484492, 73484574453020315121, 955615821857238062353, 12403944194214668554202
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2020

Keywords

Links

Crossrefs

Row 3 of A307026.
Cf. A288527, A339761.

Programs

  • Python
    # Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A(start, goal, n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal)
    return paths.len()
def A307026(n, k):
    m = k * n
    s = 0
    for i in range(1, m):
        for j in range(i + 1, m + 1):
            s += A(i, j, n, k)
    return s
def A339751(n):
    return A307026(n, 3)
print([A339751(n) for n in range(1, 21)])

Formula

Empirical g.f.: x*(3 + 142*x - 1234*x^2 + 6033*x^3 - 4437*x^4 + 1913*x^5 - 647*x^6 + 24874*x^7 + 25724*x^8 + 1737*x^9 + 10969*x^10 + 22767*x^11 + 24670*x^12 + 12330*x^13 + 1616*x^14 + 240*x^15 + 1008*x^16) / ((1 - x)^2 * (-1 + 8*x + 14*x^2 + 5*x^3 + 6*x^4)^2*(1 - 13*x - 2*x^2 + 45*x^3 - 24*x^4 - 22*x^5 + 9*x^6 + 8*x^7 - 6*x^8)). - Vaclav Kotesovec, Dec 16 2020
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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