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.

A378340 Triangle read by rows: T(n,k) is the number of n node connected achiral planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 3, 3, 4, 3, 0, 0, 2, 4, 7, 8, 7, 10, 8, 0, 0, 0, 4, 8, 15, 19, 22, 19, 29, 23, 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68, 0, 0, 0, 0, 8, 25, 47, 82, 104, 150, 175, 200, 176, 266, 215, 0, 0, 0, 0, 7, 26, 64, 123, 186, 288, 346, 488, 556, 634, 557, 844, 680
Offset: 3

Views

Author

Andrew Howroyd, Nov 25 2024

Keywords

Comments

See A378103 for illustration of initial terms. This sequence counts only those maps which have mirror symmetry.
The planar maps considered are without loops or isthmuses.
The number of edges is n + k - 1.

Examples

			Triangle begins:
n\k | 1  2  3  4   5   6   7   8   9  10  11  12  13
----+------------------------------------------------
  3 | 1;
  4 | 0, 1, 1;
  5 | 0, 1, 1, 2,  1;
  6 | 0, 0, 2, 3,  3,  4,  3;
  7 | 0, 0, 2, 4,  7,  8,  7, 10,  8;
  8 | 0, 0, 0, 4,  8, 15, 19, 22, 19, 29, 23;
  9 | 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68;
  ...

Links

Crossrefs

Row sums are A378339.
Column sums are A378341.
Antidiagonal sums are A378342.
Cf. A378103 (unsensed), A378336 (sensed), A002712.

Programs

  • PARI
    my(A=A378340rows(10)); for(i=1, #A, print(A[i])) \\ See Links for program.

Formula

T(n,2*n-5) = A002712(n-3). - Ya-Ping Lu, Dec 16 2024

A378335 Number of n-node connected sensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 2, 5, 23, 114, 675, 4357, 29692, 209933, 1531349, 11439545, 87202034, 676101375, 5319069978, 42378741675, 341403125204, 2777329727768, 22790623236250, 188475690185517, 1569588986566488, 13153980294670851, 110870960421707325, 939397206702263021
Offset: 3

Views

Author

Andrew Howroyd, Nov 23 2024

Keywords

Comments

See A377785 for illustration of initial terms. This sequence does not consider a planar map to be equivalent to its mirror image.

Links

Crossrefs

Row sums of A378336.
Cf. A262586, A377785, A378334, A378337, A378338, A378339, A378340.

Programs

A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 4, 4, 5, 4, 0, 0, 2, 6, 10, 14, 14, 18, 16, 0, 0, 0, 7, 18, 35, 49, 63, 69, 88, 78, 0, 0, 0, 5, 28, 74, 131, 204, 274, 345, 396, 489, 457, 0, 0, 0, 0, 26, 126, 304, 574, 893, 1290, 1708, 2137, 2503, 3071, 2938, 0, 0, 0, 0, 13, 159, 582, 1396, 2613, 4274, 6270, 8709, 11433, 14227, 16905, 20667, 20118
Offset: 3

Views

Author

Ya-Ping Lu, Nov 16 2024

Keywords

Comments

The planar maps considered are without loops or isthmuses.
In other words, a(n) is the number of embeddings in the plane of connected bridgeless planar simple graphs with n vertices and k triangular internal faces.
The number of edges is n + k - 1.
The nonzero terms in row n range from k = floor(n/2) through 2*n-5 and, thus, the number of nonzero terms is 2n - floor(n/2) - 4 = A001651(n-2).

Examples

			Triangle begins:
n\k        1     2     3     4     5     6     7     8     9    10    11
----     ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
3          1
4          0     1     1
5          0     1     1     2     1
6          0     0     2     4     4     5     4
7          0     0     2     6    10    14    14    18    16
8          0     0     0     7    18    35    49    63    69    88    78

Links

Crossrefs

Row sums are A377785.
Cf. A001651, A002713, A003094, A169808, A378336 (sensed), A378340 (achiral).
The final 3 terms of each row are in A002713, A005500, A005501.

Programs

Formula

T(n, 2*n-5) = A002713(n-3).
T(n,k) = (A378336(n,k) + A378340(n,k))/2.

Extensions

a(39) onwards from Andrew Howroyd, Nov 25 2024

A378338 Number of n edge connected sensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 8, 13, 29, 60, 121, 275, 604, 1332, 3035, 6866, 15623, 36110, 83421, 193114, 451851, 1057860, 2482062, 5864466, 13873957, 32891337, 78352991, 186929542, 446870480, 1071922233, 2575310764, 6198621655, 14957681872, 36148936396, 87506326488, 212249136940, 515546059746, 1254062268245
Offset: 3

Views

Author

Andrew Howroyd, Nov 23 2024

Keywords

Links

Crossrefs

Antidiagonal sums of A378336.
Cf. A378335, A378337, A378342 (achiral), A378343 (unsensed).

Programs

A378337 Number of connected sensed planar maps with an external face and n internal triangular faces.

Original entry on oeis.org

1, 2, 6, 32, 155, 1043, 7083, 51983, 393064, 3069809, 24522342, 199837021, 1655355627, 13907476161, 118275808160, 1016657106858, 8821284816805, 77181267608093, 680346219098844, 6037537131435871, 53904120217352519, 483921982054548241, 4366269495618350828, 39577142154691122910
Offset: 1

Views

Author

Andrew Howroyd, Nov 23 2024

Keywords

Crossrefs

Column sums of A378336.
Cf. A378335, A378338, A378341.

Programs

Showing 1-5 of 5 results.

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

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