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.

A377785 Number of n-node connected unsensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 2, 5, 19, 80, 407, 2403, 15570, 107366, 773660, 5746826, 43693041, 338367370, 2660629311, 21193182914, 170714892489, 1388711749891, 11395477091396, 94238431709661, 784796579069288, 6576997589788217, 55435506835745851, 469698698872829672, 3998806331800289798
Offset: 3

Views

Author

Ya-Ping Lu, Nov 06 2024

Keywords

Comments

Maps for a(3) through a(7) are illustrated in Links.
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 in which each internal face is a triangle.

Crossrefs

Row sums of A378103.
Cf. A005027 (2-connected), A342056 (3-connected), A378335 (sensed), A378339 (achiral), A378340.

Programs

Formula

a(n) = (A378335(n) + A378339(n))/2. - Andrew Howroyd, Nov 24 2024

Extensions

a(9) onwards from Andrew Howroyd, Nov 24 2024

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

Crossrefs

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

Programs

A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed 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, 5, 5, 6, 5, 0, 0, 2, 8, 13, 20, 21, 26, 24, 0, 0, 0, 10, 28, 55, 79, 104, 119, 147, 133, 0, 0, 0, 7, 45, 126, 230, 360, 491, 625, 735, 892, 846, 0, 0, 0, 0, 44, 227, 561, 1066, 1682, 2430, 3241, 4074, 4830, 5876, 5661
Offset: 3

Views

Author

Andrew Howroyd, Nov 23 2024

Keywords

Comments

See A378103 for illustration of initial terms. This sequence does not consider planar maps to be equivalent to their reflections.
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 up to orientation preserving isomorphisms.
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,  5,  5,   6,   5;
  7 | 0, 0, 2,  8, 13,  20,  21,  26,  24;
  8 | 0, 0, 0, 10, 28,  55,  79, 104, 119, 147, 133;
  9 | 0, 0, 0,  7, 45, 126, 230, 360, 491, 625, 735, 892, 846;
  ...
		

Crossrefs

Row sums are A378335.
Column sums are A378337.
Antidiagonal sums are A378338.
The final 3 terms of each row are in A002709, A002710, A002711.
Cf. A262586 (2-connected), A341923 (3-connected), A378103, (unsensed), A378340 (achiral).

Programs

  • PARI
    my(A=A378336rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code.

Formula

T(n,k) = 0 for n > 2*k + 1.
T(n,2*n-5) = A002709(n-3).
T(n,2*n-6) = A002710(n-4) for n >= 4.
T(n,2*n-7) = A002711(n-5) for n >= 5.

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

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.

Programs

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
		

Crossrefs

Programs

Showing 1-6 of 6 results.