A376001 -id:A376001 - cp's OEIS frontend

A375999 Narayana numbers (A001263), sorted, duplicates removed.

1, 3, 6, 10, 15, 20, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 175, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A001263, A006987, A024412, A376000, A376001.

from bisect import insort
from itertools import islice
def A375999_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        yield N0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0: break
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if n == 2*k-1:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A375999_list(nmax):
    return list(islice(A375999_generator(),nmax))

A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

Original entry on oeis.org

1, 6, 10, 15, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1210

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

All Narayana numbers A001263(n,k) with n != 2*k-1, are terms since A001263(n,k) = A001263(n,n+1-k). In particular, all positive triangular numbers except 3 are terms. Are there any other terms, i.e., is there a number A001263(2*k-1,k), k >= 2, that can be written as a Narayana number in another way? Any such number would also be a term of A376001.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000217, A001263, A006987, A098564, A374796, A375572, A375999, A376001.

from bisect import insort
from itertools import islice
def A376000_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        c = 0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0:
                if c >= 2: yield N0
                break
            central = n==2*k-1
            c += 2-central
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if central:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A376000_list(nmax):
    return list(islice(A376000_generator(),nmax))

cp's OEIS Frontend

A375999 Narayana numbers (A001263), sorted, duplicates removed.

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