Nathaniel Benjamin - cp's OEIS frontend

User: Nathaniel Benjamin

Nathaniel Benjamin's wiki page.

Nathaniel Benjamin has authored 3 sequences.

A278375 Edge-distinguishing chromatic number of ladder graph with 2n vertices.

Original entry on oeis.org

1, 3, 4, 5, 5, 6, 7, 7, 7, 8, 8, 8, 9, 9

Offset: 1

Nathaniel Benjamin, Nov 19 2016

O. Frank, F. Harary, and M. J. Plantholt, The line-distinguishing chromatic number of a graph, pp. 14, 241-252 of Ars Combinatoria (1982).

K. Al-Wahabi, R. Bari, F. Harary and D. Ullman, The edge-distinguishing chromatic number of paths and cycles, pp. 17-22 of Graph Theory in Memory of G. A. Dirac (Sandbjerg, 1985). Edited by L. D. Andersen et al., Annals of Discrete Mathematics, 41. North-Holland Publishing Co., Amsterdam-New York, 1989.

A275586 Numbers k that appear more than once in c_{m,n} for integers m >= n >= 1 where c_{m,n} = ((m+n)!(m-n+1))/((n)!(m+1)!).

Original entry on oeis.org

1, 2, 5, 9, 14, 20, 27, 28, 35, 42, 44, 48, 54, 65, 75, 77, 90, 104, 110, 119, 132, 135, 152, 154, 165, 170, 189, 208, 209, 230, 252, 273, 275, 297, 299, 324, 350, 377, 405, 429, 434, 440, 464, 495, 527, 544, 560, 572, 594, 629, 637, 663, 665, 702, 740, 779, 798, 819, 860, 902, 910, 945, 950, 989

Offset: 1

Edgar Jaramillo Rodriguez, Nathaniel Benjamin, Eric Jovinelly, Eugene Fiorini, Aug 02 2016

nonn

Integers that do not appear uniquely in the Catalan triangle A009766.

			The Catalan triangle (A009766) starts:
1
1, 1
1, 2, 2
1, 3, 5,  5
1, 4, 9, 14, 14
Each entry is the sum of elements in the previous row except for those which are further right. The columns are nondecreasing, and all positive integers appear in the second column.
Since 2 appears twice in the triangle, it is in the sequence. Since 6 appears only once in the triangle, it is not in the sequence. - _Michael B. Porter_, Aug 05 2016

Giovanni Resta, Table of n, a(n) for n = 1..10000
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, 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

Cf. A009766, A275481 (complement).

def remove_duplicates(values):
    output = []
    seen = set()
    for value in values:
        if value not in seen:
            output.append(value)
            seen.add(value)
    return output
def Non_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])
    non_unique = []
    for n in l:
        if  n <= k and n>1 and l.count(n) > 1:
            non_unique.append(n)
    non_unique = remove_duplicates(non_unique)
    print (non_unique)

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))

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User: Nathaniel Benjamin

A278375 Edge-distinguishing chromatic number of ladder graph with 2n vertices.

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A275586 Numbers k that appear more than once in c_{m,n} for integers m >= n >= 1 where c_{m,n} = ((m+n)!(m-n+1))/((n)!(m+1)!).

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

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Python