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

%I A376000 #11 Sep 11 2024 00:45:44
%S A376000 1,6,10,15,21,28,36,45,50,55,66,78,91,105,120,136,153,171,190,196,210,
%T A376000 231,253,276,300,325,336,351,378,406,435,465,490,496,528,540,561,595,
%U A376000 630,666,703,741,780,820,825,861,903,946,990,1035,1081,1128,1176,1210
%N A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.
%C A376000 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.
%H A376000 Pontus von Brömssen, <a href="/A376000/b376000.txt">Table of n, a(n) for n = 1..10000</a>
%o A376000 (Python)
%o A376000 from bisect import insort
%o A376000 from itertools import islice
%o A376000 def A376000_generator():
%o A376000     yield 1
%o A376000     nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
%o A376000     while 1:
%o A376000         N0 = nkN_list[0][2]
%o A376000         c = 0
%o A376000         while 1:
%o A376000             n, k, N = nkN_list[0]
%o A376000             if N > N0:
%o A376000                 if c >= 2: yield N0
%o A376000                 break
%o A376000             central = n==2*k-1
%o A376000             c += 2-central
%o A376000             del nkN_list[0]
%o A376000             insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
%o A376000             if central:
%o A376000                 insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
%o A376000 def A376000_list(nmax):
%o A376000     return list(islice(A376000_generator(),nmax))
%Y A376000 Cf. A000217, A001263, A006987, A098564, A374796, A375572, A375999, A376001.
%K A376000 nonn
%O A376000 1,2
%A A376000 _Pontus von Brömssen_, Sep 06 2024