A375999 -id:A375999 - cp's OEIS frontend

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

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))

A376001 Numbers that can be written as a Narayana number (A001263) in at least 3 ways.

Original entry on oeis.org

1, 105, 1176, 4950, 5713890

Offset: 1

Pontus von Brömssen, Sep 06 2024

The first 5 terms are triangular numbers.
a(2), ..., a(5) can all be written as a Narayana number in exactly 4 ways.
a(6) > 2*10^35 (if it exists).

			With T(n,k) = A001263(n,k):
      105 = T( 7,3) = T( 7, 5) = T(  15,2) = T(  15,  14);
     1176 = T( 9,4) = T( 9, 6) = T(  49,2) = T(  49,  48);
     4950 = T(11,4) = T(11, 8) = T( 100,2) = T( 100,  99);
  5713890 = T(92,3) = T(92,90) = T(3381,2) = T(3381,3380).

Cf. A000217, A001263, A003015, A374796, A375573, A375999, A376000.

from math import isqrt
from bisect import insort
from itertools import islice
def A010054(n):
    return isqrt(m:=8*n+1)**2 == m
def A376001_generator():
    yield 1
    nkN_list = [(5, 3, 20)] # 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 >= 3 or A010054(N0): 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 A376001_list(nmax):
    return list(islice(A376001_generator(),nmax))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python