A007401 -id:A007401 - cp's OEIS frontend

A006294 Number of rooted planar maps with n edges.

1, 1, 5, 22, 164, 1030, 8885, 65954, 614404, 5030004, 49145460, 429166584, 4331674512, 39599553708, 409230997461, 3871362876810, 40730958917220, 395684757649324, 4222043580320852, 41894315105061848, 452123832420881296

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

W. T. Tutte, On the enumeration of planar maps. Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Row maxima of A269920.

More terms from Sean A. Irvine, Feb 24 2017

A006297 Number of rooted genus-1 maps with n edges.

Original entry on oeis.org

1, 10, 167, 1720, 24164, 256116, 3392843, 36703824, 472592916, 5188948072, 65723863196, 729734918432, 9145847808784, 102432266545800, 1274461449989715, 14373136466094880, 177882700353757460, 2017523504473479992, 24868664942648145372

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Row maxima of A269921.

More terms from Sean A. Irvine, Feb 24 2017

A184336 a(n) = n + floor((3*n)^(1/3) - 2/3).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71

Offset: 1

Robert G. Wilson v, Feb 13 2011

Complement: 2, 8, 19, 37, 65, 104, 157, 224, 310, 414, 540, 689, 863, ..., .

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007401.

A184336:=n->n+floor(-(2/3) + (3*n)^(1/3)); seq(A184336(k), k=1..100); # Wesley Ivan Hurt, Nov 08 2013

f[n_] := n + Floor[-2/3 + (3 n)^(1/3)]; Array[f, 66]
Table[n+Floor[Surd[3n,3]-2/3],{n,70}] (* Harvey P. Dale, Jan 25 2023 *)

A234349 Maximal number of points that can be placed on a triangular grid of side n so that no three points are collinear.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 25, 27, 28

Offset: 1

Heinrich Ludwig, Dec 24 2013

Length of the n-th row in triangle A194136 and triangle A234350.

Differs from A007401 first at n=14.

			In a triangular grid of side 5 at most 7 points (x) can be placed so that no three of them are on a straight line. (There are exactly 2 ways to do it, rotations and reflections ignored.)
        .              x
       . x            . .
      x . x          x . x
     x . x .        . x x .
    . x . x .      . x . x .

Cf. A194136, A234350.

a(13)-a(14) from Heinrich Ludwig, Jan 10 2014
a(15)-a(16) from Heinrich Ludwig, Jan 28 2014
a(17)-a(21) from Rob Pratt, Jul 27 2015

A275481 Integers that appear uniquely in the Catalan triangle, A009766.

Original entry on oeis.org

3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Edgar Jaramillo Rodriguez, Nathaniel Benjamin, Eric Jovinelly, and Eugene Fiorini, Jul 29 2016

n appears once in c_{m,k} for integers m >= k >= 1 where c_{m,k} = ((n+k)!(n-k+1))/((k)!(n+1)!).

D. F. Bailey, Counting arrangements of 1's and-1's, Mathematics Magazine, 69 (1996): 128-131.
Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, and Tony W. H. Wong, Primes and Perfect Powers in the Catalan Triangle, J. Int. Seq., Vol. 22 (2019), Article 19.7.6.
Eric W. Weisstein, Catalan's Triangle

Subsequence of A007401, which is the complement of A000096.

Cf. A009766, A275586 (complement).

Block[{T, nn = 85}, T[n_, k_] := T[n, k] = Which[k == 0, 1, k > n, 0, True, T[n - 1, k] + T[n, k - 1]]; Rest@ Complement[Range@ nn, Union@ Flatten@ Table[T[n, k], {n, 2, nn}, {k, 2, n}]]] (* Michael De Vlieger, Feb 04 2020, after Jean-François Alcover at A009766 *)

#prints the unique integers less than k
def Unique_Catalan_Triangle(k):
    t = []
    t.append([])
    t[0].append(1)
    for h in range(1, k):
        t.append([])
        t[0].append(1)
    for i in range(1, k):
        for j in range(0, k):
            if i>j:
                t[i].append(0)
            else:
                t[i].append(t[i-1][j] + t[i][j-1])
    l = []
    for r in range(0, k):
        for s in range(0, k):
            l.append(t[r][s])
    unique = []
    for n in l:
        if n <= k and l.count(n) == 1 :
            unique.append(n)
    print(sorted(unique))

A001851 Total diameter of unlabeled trees with n nodes.

Original entry on oeis.org

0, 1, 2, 5, 9, 21, 44, 103, 232, 571, 1368, 3441, 8653, 22263, 57518, 150733, 396937, 1054540, 2815453, 7562159, 20399714, 55280532, 150356806, 410432671, 1123902587, 3086827578, 8500975970, 23470712241, 64952794778, 180144093389

Offset: 1

N. J. A. Sloane

nonn

C program around function Gen() by Gang Li & Frank Ruskey. See C program link.

			See the trees in first Bomfim link. So a(6) = 21.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. Bomfim, Diameters of the trees of order 6
W. Bomfim, C program
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010

a(21)-a(30) from Washington Bomfim, Jan 31 2011

A356375 Number of unlabeled centered trees with n nodes that have exactly one diametral path (up to direction of traversal).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 5, 9, 21, 44, 107, 247, 607, 1465, 3649, 9087, 23059, 58831, 151832, 394074, 1030492, 2708343, 7157735, 19002282, 50676945, 135691504, 364725995, 983775878, 2662271414, 7226368722, 19670528467, 53685042694, 146879757368, 402786655780, 1106968400532

Offset: 0

Geoffrey Critzer, Aug 04 2022

nonn

A diametral path in a tree is a path of maximum length. A diametral path in a centered tree is necessarily of even length. Its endpoints are leaves and its middle point is the center of the tree. A centered tree with exactly one diametral path of length 2m can be decomposed into a rooted tree of height at most m-1 along with exactly 2 rooted trees of height exactly m-1. It appears that almost all centered trees (A000676) have exactly one diametral path.

J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000676.

nn = 35; S[0, x_] := x; S[k_, x_] := Total[Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, nn}], x] &, {1}, k] Table[x^i, {i, 1, nn + 1}]]; R[0, x] := x;R[k_, x_] := S[k, x] - S[k - 1, x]; ReplacePart[ Sum[PadRight[
   CoefficientList[Series[S[m, x] (R[m, x]^2 + (R[m, x] /. x -> x^2))/2, {x, 0, nn}],x], nn + 1], {m, 0, nn/2}], 2 -> 1]

cp's OEIS Frontend

A006294 Number of rooted planar maps with n edges.

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A006297 Number of rooted genus-1 maps with n edges.

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A184336 a(n) = n + floor((3*n)^(1/3) - 2/3).

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Maple

Mathematica

A234349 Maximal number of points that can be placed on a triangular grid of side n so that no three points are collinear.

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Extensions

A275481 Integers that appear uniquely in the Catalan triangle, A009766.

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Mathematica

Python

A001851 Total diameter of unlabeled trees with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Extensions

A356375 Number of unlabeled centered trees with n nodes that have exactly one diametral path (up to direction of traversal).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica