A378340 -id:A378340 - cp's OEIS frontend

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

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

Ya-Ping Lu, Nov 06 2024

nonn

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.

Andrew Howroyd, Table of n, a(n) for n = 3..200
Ya-Ping Lu, Illustration of initial terms

Row sums of A378103.

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

A377785seq(30) \\ See PARI link in A378340 for program code.

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

a(9) onwards from Andrew Howroyd, Nov 24 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

Andrew Howroyd, Nov 23 2024

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 3..200

Row sums of A378336.

Cf. A262586, A377785, A378334, A378337, A378338, A378339, A378340.

A378335seq(25) \\ See PARI link in A378340 for program code.

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

Andrew Howroyd, Nov 24 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..200

Row sums of A378340.

Cf. A377785 (unsensed), A378335 (sensed), A378341 (by faces), A378342 (by edges).

A378339seq(30) \\ See PARI link in A378340 for program code.

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

Ya-Ping Lu, Nov 16 2024

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).

			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

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
Ya-Ping Lu, Illustration of initial terms

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.

my(A=A378103rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code. - Andrew Howroyd, Nov 25 2024

T(n, 2*n-5) = A002713(n-3).

T(n,k) = (A378336(n,k) + A378340(n,k))/2.

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

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

Andrew Howroyd, Nov 23 2024

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.

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)

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).

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

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

Andrew Howroyd, Nov 23 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100

Antidiagonal sums of A378336.

Cf. A378335, A378337, A378342 (achiral), A378343 (unsensed).

A378338seq(30) \\ See PARI link in A378340 for program code.

A378342 Number of n edge connected achiral planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 6, 7, 15, 22, 33, 59, 94, 140, 249, 388, 617, 1040, 1675, 2628, 4481, 7108, 11494, 19030, 30885, 49315, 82559, 132406, 215326, 353923, 576552, 928437, 1539504, 2485102, 4051828, 6636368, 10830526, 17524893, 28903185, 46838466, 76464837, 125019717

Offset: 3

Andrew Howroyd, Nov 25 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100

Antidiagonal sums of A378340.

Cf. A378338 (sensed), A378339, A378341, A378343 (unsensed).

A378338seq(45) \\ See PARI link in A378340 for program code.

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

Andrew Howroyd, Nov 23 2024

nonn

Column sums of A378336.

Cf. A378335, A378338, A378341.

A378337seq(20) \\ See PARI link in A378340 for program code.

A378341 Number of connected achiral planar maps with an external face and n internal triangular faces.

Original entry on oeis.org

1, 2, 6, 16, 45, 129, 367, 1081, 3098, 9291, 26788, 81341, 235729, 722479, 2102856, 6492146, 18965527, 58898531, 172600126, 538640825, 1582733185, 4959648919, 14607576158, 45935431662, 135571479679, 427621459608, 1264356405112, 3998682343629, 11842126315245, 37540318975749

Offset: 1

Andrew Howroyd, Nov 25 2024

nonn

Column sums of A378340.

Cf. A378337, A378339, A378342.

A378341seq(30) \\ See PARI link in A378340 for program code.

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

Andrew Howroyd, Nov 25 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100

Antidiagonal sums of A378103.

Cf. A377785 (by vertices), A378338 (sensed), A378342 (achiral).

A378343seq(30) \\ See PARI link in A378340 for program code.

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

cp's OEIS Frontend

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

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

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

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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.

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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.

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

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A378342 Number of n edge connected achiral planar maps with an external face in which all internal faces are triangles.

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

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A378341 Number of connected achiral planar maps with an external face and n internal triangular faces.

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