A375573 -id:A375573 - cp's OEIS frontend

A375572 Numbers occurring at least twice in Bernoulli's triangle A008949.

1, 4, 7, 8, 11, 15, 16, 22, 26, 29, 31, 32, 37, 42, 46, 56, 57, 63, 64, 67, 79, 92, 93, 99, 106, 120, 121, 127, 128, 130, 137, 154, 163, 172, 176, 191, 211, 219, 232, 247, 254, 255, 256, 277, 299, 301, 326, 352, 378, 379, 382, 386, 407, 436, 466, 470, 497, 502

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Equivalently, 1 together with numbers occurring in columns k >= 2 of Bernoulli's triangle.

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

Cf. A006987, A008949, A375570, A375573.

isok(k) = my(nb=0); for (i=0, k, nb += #select(x->(x==k), vector(i+1, j, sum(jj=0, j-1, binomial(i, jj))))); nb >= 2; \\ Michel Marcus, Aug 22 2024

lista(nn) = my(v = vector(nn)); for (n=1, nn, my(w=vector(n+1, j, sum(jj=0, j-1, binomial(n, jj)))); for (i=1, #w, if (w[i] <= nn, v[w[i]]++));); Vec(select(x->(x>=2), v, 1)); \\ Michel Marcus, Aug 23 2024

from math import comb
from bisect import insort
def A375572_list(nmax):
    a_list = [1]
    if nmax == 1: return a_list
    nkb_list = [(2,2,4)] # List of triples (n,k,A008949(n,k)), sorted by the last element.
    while 1:
        b0 = nkb_list[0][2]
        a_list.append(b0)
        if len(a_list) == nmax: return a_list
        while 1:
            n,k,b = nkb_list[0]
            if b > b0: break
            del nkb_list[0]
            insort(nkb_list,(n+1,k,2*b-comb(n,k)),key=lambda x:x[2])
            if n == k:
                insort(nkb_list,(n+1,k+1,2**(k+1)),key=lambda x:x[2])

A375570 Smallest m such that A008949(m,k) = n for some k.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 3, 3, 8, 9, 4, 11, 12, 13, 4, 4, 16, 17, 18, 19, 20, 6, 22, 23, 24, 5, 26, 27, 7, 29, 5, 5, 32, 33, 34, 35, 8, 37, 38, 39, 40, 6, 42, 43, 44, 9, 46, 47, 48, 49, 50, 51, 52, 53, 54, 10, 6, 57, 58, 59, 60, 61, 6, 6, 64, 65, 11, 67, 68, 69, 70

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Cf. A008949, A058084, A375571, A375572, A375573.

A008949(a(n),A375571(n)) = n.
a(n) <= n-1.
a(2^n) = n.
a(2^n-1) = n for n >= 2.

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

A375572 Numbers occurring at least twice in Bernoulli's triangle A008949.

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A375570 Smallest m such that A008949(m,k) = n for some k.

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

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