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%I #19 Feb 07 2014 10:16:27
%S 0,6,3,15,741,153,25878,3828,16653,253,4753,23653,6328,53956,9730,
%T 191890,21115,140185,1225,26335,317206,3160,38503,2108431,37950,
%U 121278,1440905403,53483653,201733741,58595725,22663524351,787786971,23483020686,1475521326
%N Sequence of distinct least triangular numbers such that the arithmetic mean of the first n terms is also a triangular number. Initial term is 0.
%C Sequence is believed to be infinite. If a(31) exists, it will be greater than 10^10.
%C a(31)=22663524351. - _Jon E. Schoenfield_, Feb 07 2014
%e a(1) = 0.
%e a(2) must be a triangular number such that (a(1)+a(2))/2 is triangular. Thus, a(1) = 6.
%e a(3) must be a triangular number such that (a(1)+a(2)+a(3))/3 is triangular. Thus, a(3) = 3.
%e ...and so on
%o (Python)
%o def Tri(x):
%o ..for n in range(10**10):
%o ....if x == n*(n+1)/2:
%o ......return True
%o ....if x < n*(n+1)/2:
%o ......return False
%o ..return False
%o def TriAve(init):
%o ..print(init)
%o ..lst = []
%o ..lst.append(init)
%o ..n = 1
%o ..while n*(n+1)/2 < 10**10:
%o ....if n*(n+1)/2 not in lst:
%o ......if Tri(((sum(lst)+int(n*(n+1)/2))/(len(lst)+1))):
%o ........print(int(n*(n+1)/2))
%o ........lst.append(int(n*(n+1)/2))
%o ........n = 1
%o ......else:
%o ........n += 1
%o ....else:
%o ......n += 1
%Y Cf. A000217.
%K nonn
%O 1,2
%A _Derek Orr_, Jan 25 2014
%E More terms from _Jon E. Schoenfield_, Feb 07 2014
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