A378337 -id:A378337 - cp's OEIS frontend

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

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.

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.

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.

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

Ya-Ping Lu, Nov 19 2024

nonn

Column sums of A378103.

Cf. A002713, A169808, A377785.

a(n) = (A378337(n) + A378341(n))/2.

cp's OEIS Frontend

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

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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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PARI

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

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

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