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 1-6 of 6 results.

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

A378339 Number of n-node connected achiral planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 2, 5, 15, 46, 139, 449, 1448, 4799, 15971, 54107, 184048, 633365, 2188644, 7624153, 26659774, 93772014, 330946542, 1173233805, 4171572088, 14884905583, 53249784377, 191043396323, 686956509123, 2476044740337, 8942288625630, 32360306616473, 117309644377341
Offset: 3

Views

Author

Andrew Howroyd, Nov 24 2024

Keywords

Links

Crossrefs

Row sums of A378340.
Cf. A377785 (unsensed), A378335 (sensed), A378341 (by faces), A378342 (by edges).

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

A378343 Number of n edge connected unsensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 7, 10, 22, 41, 77, 167, 349, 736, 1642, 3627, 8120, 18575, 42548, 97871, 228166, 532484, 1246778, 2941748, 6952421, 16470326, 39217775, 93530974, 223542903, 536138078, 1287943658, 3099775046, 7479610688, 18075710749, 43755189158, 106127886654, 257778445136
Offset: 3

Views

Author

Andrew Howroyd, Nov 25 2024

Keywords

Links

Crossrefs

Antidiagonal sums of A378103.
Cf. A377785 (by vertices), A378338 (sensed), A378342 (achiral).

Programs

Formula

a(n) = (A378338(n) + A378342(n))/2.

A378334 Number of n-node connected sensed planar maps with an external face, whose internal faces are triangles and which is rooted at a node on the external face.

Original entry on oeis.org

1, 3, 16, 88, 537, 3473, 23562, 165859, 1202919, 8941118, 67836666, 523722889, 4104244351, 32583588994, 261635413752, 2122002356439, 17364496928654, 143230857011191, 1189928600338968, 9949848535116510, 83688384576105358, 707689149415139246, 6013842677410501394
Offset: 3

Views

Author

Andrew Howroyd, Nov 23 2024

Keywords

Examples

			The a(4) = 3 rooted planar maps are:
    R---o    R---o      R-----o
    | \ |    | / |      | \ / |
    o---o    o---o       \ o /
                          \|/
                           o

Links

Crossrefs

Cf. A377785, A378335, A378339, A378340.

Programs

A378190 Number of planar maps with an external face and n internal triangular faces.

Original entry on oeis.org

1, 2, 6, 24, 100, 586, 3725, 26532, 198081, 1539550, 12274565, 99959181, 827795678, 6954099320, 59138955508, 508331799502, 4410651891166, 38590663253312, 340173195849485, 3018768835038348, 26952060900042852, 241960993507098580, 2183134755112963493, 19788571100313277286
Offset: 1

Views

Author

Ya-Ping Lu, Nov 19 2024

Keywords

Crossrefs

Column sums of A378103.
Cf. A002713, A169808, A377785.

Formula

a(n) = (A378337(n) + A378341(n))/2.
Showing 1-6 of 6 results.

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

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