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

Author

Pontus von Brömssen, Aug 19 2024

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A006987, A008949, A375570, A375573.

Programs

PARI

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

PARI

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

Python

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])