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

1,2

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

Prog

(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])

Crossrefs

Cf. A006987, A008949, A375570, A375573.

Sequence in context: A137362 A190765 A024621 * A296029 A000606 A215458

Adjacent sequences: A375569 A375570 A375571 * A375573 A375576 A375577

Keyword

nonn,new

Author

Pontus von Brömssen, Aug 19 2024

Status

approved