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A375572
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Numbers occurring at least twice in Bernoulli's triangle A008949.
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2
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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
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OFFSET
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1,2
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COMMENTS
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Equivalently, 1 together with numbers occurring in columns k >= 2 of Bernoulli's triangle.
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LINKS
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PROG
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(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
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])
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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